Question

If a blue photon having \(3.3\) electron-volt of energy liberates an electron in silicon, whose band gap is \(1.1 \mathrm{eV}_{\prime \prime}\), what fraction of the photon's energy is "kept" by the electron once it settles down from the excess?

Short answer

Answer: 66.67%

Step-by-step solution

Identify the given values

We are given the energy of the blue photon (E_photon) as 3.3 eV and the band gap of silicon (E_gap) as 1.1 eV.

Calculate energy retained by the electron after liberation

The energy retained by the electron (E_electron) is the difference between the blue photon's energy and the silicon's band gap energy. Thus, E_electron = E_photon - E_gap. E_electron = 3.3 eV - 1.1 eV = 2.2 eV

Find the ratio of energy retained by the electron to the initial energy of the photon

Now, we will find the fraction of energy kept by the electron by dividing E_electron by E_photon. Fraction = \(\frac{E_{electron}}{E_{photon}}\) = \(\frac{2.2 \: \text{eV}}{3.3 \: \text{eV}}\) = \(\frac{2}{3}\)

Express the fraction as a percentage

To express the fraction as a percentage, multiply the fraction by 100. Percentage = \(\frac{2}{3} \times 100\) = 66.67% So, 66.67% of the blue photon's energy is "kept" by the electron once it settles down from the excess.

Key concepts

Band Gap Energy

Band gap energy is a fundamental concept in solid-state physics, particularly concerning semiconductors like silicon. The term 'band gap' refers to the energy difference between the top of the valence band and the bottom of the conduction band in a crystalline material. Electrons within any material can only exist at specific energy levels, and the band gap is the energy region where no electron states can exist.

In semiconductors, the band gap determines which photons (light particles) can be absorbed to create free charge carriers, such as electrons in the conduction band. For silicon, the band gap is approximately 1.1 electronvolts (eV). This value means that any photon with energy greater than 1.1 eV can excite an electron from the valence band to the conduction band, giving it enough energy to jump across the band gap.

• Silicon's band gap is instrumental in its application in photovoltaics and electronics.
• The ability of silicon to absorb light and generate electricity is central to how solar panels work.

Electron Energy Retention

When a photon interacts with silicon and generates a pair of charge carriers (an electron and a hole), the electron retains a portion of the photon's energy. After absorption, the energy of the photon is partially used to bridge the band gap, and the rest contributes to the kinetic energy of the generated electron-holes pair.

Here's what happens step by step:

1. A photon with energy greater than the band gap energy is absorbed by silicon.
2. This energy liberates an electron from its atomic structure, allowing it to move freely in the conduction band.
3. The electron retains the excess energy after the band gap energy is considered.

In our exercise, an electron absorbs a photon of 3.3 eV and uses 1.1 eV to cross the band gap, retaining the difference in energy which is not required for liberation.

Blue Photon

Photons are elementary particles that carry electromagnetic radiation, including visible light. The energy of a photon is determined by its wavelength, with blue photons being on the higher energy end of the visible spectrum.

The energy of a photon can be calculated using the equation:
\( E = h \times f \)
where:

• \( E \) is the energy of the photon,
• \( h \) is the Planck constant,
• \( f \) is the frequency of the photon.

For a blue photon, with typical energies around 3 eV to 3.3 eV, we observe that their energy is sufficient to excite electrons in silicon, given its band gap of approximately 1.1 eV. This feature explains why silicon is a common material in devices intended to detect or harvest light, such as cameras and solar cells.

Energy Conservation in Physics

Energy conservation is a fundamental principle in physics stating that the total energy in a closed system remains constant over time, irrespective of the changes that occur within the system. In the context of our exercise, when a photon interacts with an electron in silicon, the energy before and after the interaction must balance according to the law of conservation of energy.

For the blue photon with 3.3 eV of energy, its initial energy is partially used to excite the electron across the band gap, and the rest contributes to the electron's kinetic energy. By calculating the fraction of energy retained by the electron, we essentially observe energy conservation at play:

• The initial energy of the photon is split between overcoming the band gap and increasing the electron's energy.
• No energy is lost; instead, it is transformed from one form to another, maintaining the principle of energy conservation.