Problem 19 - University Physics with Modern 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.

Step-by-Step Solution

Answer

(a) The energy gap is approximately 1.12 eV. (b) Silicon is opaque because it absorbs photons with energies equal to or above its band gap.

Step 1: Understand the relationship between wavelength and energy

The energy of a photon is related to its wavelength by the formula \( E = \frac{hc}{\lambda} \), where \( h \) is Planck's constant (approximately \( 4.1357 \times 10^{-15} \text{ eV s} \)), \( c \) is the speed of light (\( 3 \times 10^8 \text{ m/s} \)), and \( \lambda \) is the wavelength.

Step 2: Identify the given wavelength

For this exercise, the maximum wavelength \( \lambda \) is given as 1.11 micrometers (\( 1.11 \times 10^{-6} \text{ m} \)). Convert this into meters if necessary for calculation purposes.

Step 3: Calculate the energy in electron volts

Use the formula \( E = \frac{hc}{\lambda} \) to calculate the energy. Substitute \( h \), \( c \), and \( \lambda \) into the formula:\[E = \frac{4.1357 \times 10^{-15} \times 3 \times 10^8}{1.11 \times 10^{-6}}\]Calculate this to find \( E \) in electron volts.

Step 4: Perform the calculation

Compute the above expression:\[E \approx \frac{4.1357 \times 10^{-15} \times 3 \times 10^8}{1.11 \times 10^{-6}} = 1.12 \text{ eV}\]Thus, the energy gap \( E_g \) is approximately 1.12 eV.

Step 5: Explain why pure silicon is opaque

Pure silicon is opaque because its energy band gap (around 1.12 eV) allows absorption of photons with energies equal to or greater than the band gap. This means that photons with such energies can be absorbed to excite electrons from the valence to the conduction band, thus causing the material to absorb rather than transmit light.

Key Concepts

Photon Energy and Wavelength

The energy of a photon is directly linked to its wavelength through a fundamental equation in physics: \( E = \frac{hc}{\lambda} \). This equation shows that the energy \( E \) of a photon is inversely proportional to its wavelength \( \lambda \).
Here, \( h \) represents Planck's constant, and \( c \) is the constant speed of light in a vacuum. As the wavelength increases, the energy of the photon decreases, and vice versa.

• Shorter wavelengths correspond to higher energy photons, like ultraviolet light.
• Longer wavelengths indicate lower energy, such as infrared light.

Understanding this relationship is crucial when dealing with electronics that involve light detection or light emission, such as silicon photocells. When calculating the energy a photon can carry, it's essential to convert the wavelength into meters, especially when dealing with micrometers, to match the units used in the constants.

Planck's Constant

Planck's constant, denoted as \( h \), is one of the cornerstones of quantum mechanics and plays a vital role in understanding photon energy. Its value is approximately \( 4.1357 \times 10^{-15} \text{ eV s} \). This constant helps link the energy of a photon to its frequency, another way to express how energy and light relate.
Planck's constant appears in various formulas, highlighting its critical role in quantum physics:

• For energy related to frequency, \( E = hf \), where \( f \) is frequency.
• For wavelength relationships, it forms part of \( E = \frac{hc}{\lambda} \).

These relationships allow us to convert between different characteristics of electromagnetic waves, like energy and wavelength. Knowing Planck's constant is crucial for solving problems where light and matter interact.

Energy Band Gap of Silicon

The energy band gap is a fundamental property that defines the conductance of materials like silicon. For semiconductors, the energy band gap stands between the valence and conduction bands. Electrons need a certain amount of energy to jump this gap and contribute to electrical conductivity.
For silicon, this band gap is approximately 1.12 eV. This energy threshold determines whether a photon will be absorbed and cause an electron to move from the valence to the conduction band.

• Photons with energy equal to or greater than the band gap will be absorbed, triggering electronic activity.
• Lower energy photons will pass through without absorption, making them invisible to the silicon material.

This principle explains why pure silicon can appear opaque, as it can absorb photons with sufficient energy to stimulate an electron transition, effectively capturing that light rather than letting it pass through. Understanding the energy band gap is key to designing and understanding semiconductors in electronic devices.