Question

The resivivity of silver is \(1.6 \times 10^{-8} \Omega\) - m at room temperature \((300 \mathrm{~K}) .\) while that of silicon is about \(10 \Omega \cdot m\) (a) Show that ths disparity follows, at least to a rough order of magnitude, from the approximate I \(\mathrm{eV}\) band gap in silicon. (b) What would you expect for the noom-temperature resistivity of diamond, which has a band gap of about \(5 \mathrm{eV?}\)

Short answer

By understanding resistivity and its relationship with energy gap, it was found that the major difference in resistivity of silver (a conductor) and silicon (a semiconductor) is due to the difference in their energy gaps. Silver has virtually no energy gap, leading to lower resistivity, while silicon has a band gap energy of about 1 eV, leading to higher resistivity. The resistivity of diamond, with a band gap energy of about 5 eV, would be expected to be significantly higher than that of silicon.

Step-by-step solution

Understand Resistivity and Energy Gap Relationship

It's crucial to note that resistivity, symbolized by \(\rho\), is directly related to the band gap energy of materials. The band gap energy, denoted by \(E_g\), of a material relates to the energy required to make an electron jump from the valence band to the conduction band where it's free to conduct electric current. In other words, the higher the band gap, the higher the resistivity, hence more energy is needed to make an electron conduction-capable.

Compare Resistivity of Silver and Silicon

For silver, which is a conductor, the resistivity is extremely low. This is because it has virtually no band gap, meaning electrons are free to move to the conduction band. On the other hand, silicon is a semiconductor, and has a band gap energy of about 1 eV. Therefore, it requires substantial energy for electrons to move to conduction band, resulting in higher resistivity.

Calculate Rough Order of Magnitude Difference

Comparing the resistivites, the disparity is roughly of the order \(10\Omega.m / 1.6 \times 10^{-8} \Omega.m = 6.25 \times 10^{8}\). Since silicon's bandgap E is about 1 eV, each order of magnitude in resistivity difference should correlate roughly with about an order of magnitude difference in E. Hence, it matches up -- the bands of magnitude difference seen in resistivity match with the fact that a semiconductor such as silicon has a finite band gap while a metal doesn't.

Predict Resitivity of Diamond

Diamond has a much larger bandgap energy of about 5 eV. Since the resistivity is directly related to the band gap energy, it's reasonable to expect that diamond's resistivity at room temperature is significantly higher than that of silicon by order of magnitude difference in E.

Key concepts

Resistivity in Materials

Understanding resistivity in materials is pivotal when exploring their electrical properties. Resistivity, noted as \( \rho \), is a measure of how strongly a material opposes the flow of electric current. It's a fundamental characteristic that differentiates conductors from insulators. Conductors, such as silver, exhibit very low resistivity because of their ability to let electrons flow freely through them. In contrast, insulators have high resistivity, making them poor conductors of electricity.

The resistivity of a material is not just about the presence of free electrons; it is also greatly influenced by the material's temperature, structure, and the presence of impurities. These factors can alter resistivity significantly. For instance, increasing temperatures typically increase the resistivity of conductors while potentially decreasing that of semiconductors—up to a certain point. Material purity can also have a pronounced effect, with higher impurity levels generally increasing resistivity.

Semiconductor Band Gap

The semiconductor band gap is a term that carries critical importance in the field of electronics. It represents the energy difference between the valence band, filled with electrons, and the conduction band, where electrons can move freely and conduct electricity. Semiconductors, like silicon, have a band gap that allows them to act as conductors or insulators depending on temperature, impurities, and external stimuli.

The size of the band gap has direct implications on a material's electrical properties. For instance, the band gap of silicon is approximately 1 electronvolt (eV), which is sufficient to prevent free electron flow at room temperature but allows conduction at higher temperatures or with doping—introduction of impurities. Materials with larger band gaps, like diamond with a band gap around 5 eV, show significantly higher resistivity at room temperature, as more energy is required to excite an electron into the conduction band.

Conductors and Semiconductors

Distinguishing between conductors and semiconductors is essential when discussing electronic materials. Conductors like silver have negligible resistance to the flow of electricity due to the absence of a band gap, meaning electrons can move freely at virtually any temperature. This characteristic makes them ideal for applications requiring efficient electrical transmission.

On the other hand, semiconductors like silicon stand in the middle ground between conductors and insulators. Their conductivity can be influenced by temperature, light, or additional impurities, which makes them highly valuable for creating electronic devices like transistors, diodes, and solar cells. In practice, semiconductors can be tweaked to exhibit various levels of conductivity, yielding immense control for tech design and functionality.

Energy Band Theory

At the core of understanding electrical properties in materials is the energy band theory. This theory explains that electrons in any material are organized in energy bands: the valence band and the conduction band, where the valence band is fully occupied by electrons and the conduction band is the next energy level that electrons can jump to for conduction.

For conductors, the valence and conduction bands overlap, meaning electrons can easily transition to conducting energy. In insulators, there's a large gap, making electron flow highly unlikely under normal conditions. Semiconductors possess a band gap that is narrower than insulators but still present, ensuring only certain electrons can make the jump. With the right conditions, such as doping or increased temperature, the conductivity of semiconductors can be enhanced. This versatile nature stems from their unique band structure, positioning semiconductors as critical components in modern electronics.