Question

The maximum wavelength of light that a certain silicon photocell can detect is 1.11 $\mu$m. (a) What is the energy gap (in electron volts) between the valence and conduction bands for this photocell? (b) Explain why pure silicon is opaque.

Short answer

(a) The band gap is approximately 1.12 eV. (b) Pure silicon is opaque due to a large band gap.

Step-by-step solution

Understand the Problem

We need to find the energy gap (band gap) of silicon for a given maximum wavelength that it can detect. We also need to explain why pure silicon is opaque.

Use the Energy-Wavelength Relationship

The relationship between the energy of a photon and its wavelength is given by the equation: $E=\frac{hc}{\lambda }$, where $E$ is the energy, $h$ is Planck's constant $6.626\times {10}^{-34}\text{ Js}$, $c$ is the speed of light $3\times {10}^8\text{ m/s}$, and $\lambda$ is the wavelength of the light.

Convert Wavelength to Meters

Since the given wavelength is 1.11 $\mu$m, we need to convert it to meters: $1.11\times {10}^{-6}$ meters.

Calculate Energy in Joules

Substitute the values into the energy-wavelength relationship: $E=\frac{6.626\times {10}^{-34}\times 3\times {10}^8}{1.11\times {10}^{-6}}$. Calculate $E$ in Joules.

Convert Energy to Electron Volts

To convert the energy from Joules to electron volts, use the conversion factor: $1\text{ eV}=1.602\times {10}^{-19}\text{ J}$. Thus, $E_{eV}=\frac{E}{1.602\times {10}^{-19}}$.

Explain Silicon's Opacity

Silicon is typically opaque because its band gap is large enough to prevent lower-energy photons from passing through, especially in pure form. The electrons need to be excited across this gap to conduct electricity or transmit light.

Key concepts

Silicon Photocell

Silicon photocells, often found in solar panels and light sensors, are essential devices in converting light into electricity. These devices function based on the principle of the photoelectric effect. When light hits the surface of a silicon photocell, it can knock electrons loose, creating an electric current. This phenomenon is possible due to silicon having a specific energy gap or band gap.
This energy gap is crucial because it defines the amount of energy required to free an electron from silicon's valence band to its conduction band. When light with sufficient energy (or short enough wavelength) strikes the silicon, it excites electrons into the conduction band, thereby generating electricity.
If the band gap is too large, most of the light that hits the silicon can be reflected or absorbed without causing an electrical current, as the photons do not have enough energy to move the electrons across the gap. In this exercise, the silicon photocell can detect a maximum wavelength of 1.11 µm, meaning any light with a longer wavelength does not have enough energy to affect the electrons.

Energy-Wavelength Relationship

To understand how light energy interacts with materials like silicon, it's essential to grasp the energy-wavelength relationship. Light behaves both as a wave and as particles called photons, where each photon carries energy proportional to its frequency. This relationship is governed by the equation: $$E=\frac{hc}{\lambda }$$where:

• $E$ is the energy of the photon,
• $h$ is Planck's constant $6.626\times {10}^{-34}\text{ Js}$,
• $c$ is the speed of light $3\times {10}^8\text{ m/s}$,
• $\lambda$ is the wavelength of the light.

To find the energy of a photon from its wavelength, as we did in the exercise, you can rearrange this relationship by inserting the known constants and the wavelength in meters.
For the given wavelength of 1.11 µm, which is equivalent to $1.11\times {10}^{-6}$ meters, the energy calculated is in Joules. Converting this energy to electron volts gives a value that more easily relates to the silicon's band gap. This conversion helps us understand if the light can excite electrons sufficiently in the silicon photocell.

Opacity of Silicon

Understanding why silicon can act as an opaque material, especially in its pure form, involves considering its band gap size. The band gap is the energy difference between the valence band and conduction band of the material. In solid-state physics, this gap is a measure of a substance's ability to conduct electricity in response to an electric field.
Pure silicon, due to its specific band gap of about 1.1 eV, is usually opaque to light with energy lower than its band gap energy. This means that photons with less energy cannot excite electrons across the band gap, making the silicon impermeable to these lower energy light waves.
When silicon is doped with other materials, this opacity can change, allowing it to absorb and transmit different wavelengths of light, which is why silicon is widely used in electronic devices and solar cells. Doping adjusts the band gap and, therefore, the material's translucency and electrical conductivity, enabling it to effectively convert light into electricity.