Problem 33 - College Physics

Question

The maximum wavelength of light that a certain silicon photocell can detect is 1.11\(\mu \mathrm { m }\) . (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. (Hint: Will visible light that strikes silicon be transmitted or absorbed?

Step-by-Step Solution

Answer

(a) The energy gap is 1.12 eV. (b) Pure silicon is opaque because visible light is absorbed due to its small energy gap.

Step 1: Understanding the Relationship

The energy gap, or band gap, between the valence and conduction bands can be calculated using the formula \( E = \frac{hc}{\lambda} \), 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 in meters. We need to convert the wavelength from micrometers to meters.

Step 2: Converting Wavelength

The given maximum wavelength is 1.11 micrometers. Convert this to meters by remembering that 1 micrometer is equal to \( 1 \times 10^{-6} \) meters. Thus, \( \lambda = 1.11 \times 10^{-6} \text{ m} \).

Step 3: Calculating the Energy Gap

Plug the values into the equation \( E = \frac{hc}{\lambda} \). Substitute \( h = 6.626 \times 10^{-34} \text{ Js} \), \( c = 3 \times 10^8 \text{ m/s} \), and \( \lambda = 1.11 \times 10^{-6} \text{ m} \). This gives \( E = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{1.11 \times 10^{-6}} \approx 1.79 \times 10^{-19} \text{ J} \).

Step 4: Conversion to Electron Volts

Since energy is often expressed in electron volts (eV) for such problems, convert joules to electron volts using the conversion factor \( 1 ext{ eV} = 1.602 \times 10^{-19} ext{ J} \). Therefore, \( E = \frac{1.79 \times 10^{-19} \text{ J}}{1.602 \times 10^{-19} \text{ J/eV}} \approx 1.12 \text{ eV} \).

Step 5: Explaining Silicon's Opaqueness

Silicon is opaque to light primarily because its band gap \( (1.12 \text{ eV}) \) is smaller than the energy of visible light photons (1.65 - 3.1 eV). Therefore, visible light striking silicon is absorbed rather than transmitted, making the material opaque to visible light.

Key Concepts

Silicon Photocell

A silicon photocell is a device that converts light into electrical energy using silicon, a semiconductor material. The magic happens in the light-sensitive part of the device, which is often a thin layer of silicon.
When light hits the photocell, it provides energy to the electrons in the silicon, pushing them from the valence band to the conduction band, creating an electric current.

• The effectiveness of a silicon photocell is determined by its ability to capture light and convert it to electricity.
• Silicon photocells are widely used due to their stability and efficiency. They are commonplace in solar panels, where they harness sunlight to produce power.

The process is quite similar to how plant leaves capture sunlight to make food, but instead of food, the photocells produce electricity.

Band Gap

The band gap is a fundamental concept in semiconductors, serving as the key to understanding how materials like silicon can conduct electricity.
It refers to the energy required to move an electron from the valence band, where electrons typically reside, to the conduction band where they can move freely and conduct electricity.

• The size of the band gap influences the electrical conductivity of a material. Smaller band gaps make it easier for electrons to transition between bands, enabling conduction.
• In silicon photocells, the band gap is crucial for determining which wavelengths of light the cell can absorb and convert into electrical energy.

For silicon, the band gap is approximately 1.12 electron volts, allowing it to absorb light with wavelengths up to around 1.11 micrometers.

Electron Volts

Electron volts (eV) are a convenient unit of energy used in physics to express small amounts of energy.
One electron volt is defined as the amount of kinetic energy gained or lost by an electron when it moves through an electric potential difference of one volt.

• Because typical energy calculations in atomic and subatomic physics involve minuscule energies, electron volts provide a more manageable scale.
• When working with band gaps or energy levels in semiconductors like silicon, expressing energy in electron volts instead of joules simplifies calculations and presentations of data.

In the context of silicon photocells, saying the band gap is 1.12 eV is much clearer than using the joules, as it matches the scale of the materials' properties.

Opaque Materials

Opaque materials do not allow light to pass through them. Instead, they absorb or reflect incoming light.
Silicon, in its pure form, is considered opaque under visible light. This is due to its band gap being smaller than the energy of visible light photons.

• Visible light has photon energies ranging from approximately 1.65 to 3.1 electron volts. Silicon's band gap of 1.12 eV means that it can absorb photons from the visible spectrum.
• Once absorbed, the energy promotes electrons into the conduction band, but this energy is not re-emitted as transmitted light, rendering silicon opaque.

Opaqueness in materials like silicon is a crucial characteristic that influences their applications in electronics, especially in components that rely on light absorption.