Question

`A pure, defect-free semiconductor material will absorb the electromagnetic radiation incident on that material only if the energy of the individual photons in the incident beam is larger than a threshold value known as the "band-gap" of the semiconductor. The known room-temperature band-gaps for germanium, silicon, and gallium-arsenide, three widely used semiconductors, are \(0.66 \mathrm{eV}, 1.12 \mathrm{eV},\) and \(1.42 \mathrm{eV},\) respectively. a) Determine the room-temperature transparency range of these semiconductors. b) Compare these with the transparency range of \(\mathrm{ZnSe}, \mathrm{a}\) semiconductor with a band-gap of \(2.67 \mathrm{eV},\) and explain the yellow color observed experimentally for the ZnSe crystals. c) Which of these materials could be used for a light detector for the 1550 -nm optical communications wavelength?

Short answer

Question: Calculate the minimum wavelength for germanium, silicon, and gallium-arsenide using their respective band-gap energies and compare them with the transparency range of ZnSe. Explain why ZnSe crystals appear yellow. Determine which semiconductor material is suitable for a light detector for the 1550-nm optical communications wavelength.

Step-by-step solution

Find the minimum wavelength for germanium, silicon, and gallium-arsenide using their band-gap energies

Use the energy-wavelength relationship formula to calculate the minimum wavelength for each semiconductor using their respective band-gap energies. The values of h and c are: $$ h = 6.626 \times 10^{-34} \mathrm{J} \cdot \mathrm{s}, \quad c = 3 \times 10^8 \mathrm{m}/\mathrm{s} $$ For germanium (Ge): $$ 0.66\ \mathrm{eV} =\frac{hc}{\lambda_\mathrm{Ge}} $$ Solve for \(\lambda_\mathrm{Ge}\): $$ \lambda_\mathrm{Ge} = \frac{hc}{0.66\ \mathrm{eV}} $$ and similar for silicon (Si) and gallium-arsenide (GaAs): $$ \lambda_\mathrm{Si} = \frac{hc}{1.12\ \mathrm{eV}}, \quad \lambda_\mathrm{GaAs} = \frac{hc}{1.42\ \mathrm{eV}} $$

Compare transparency ranges with ZnSe and explain the yellow color

Find the minimum wavelength for ZnSe using the band-gap: $$ \lambda_\mathrm{ZnSe} = \frac{hc}{2.67\ \mathrm{eV}} $$ Compare the transparency ranges of the semiconductors and relate them to the visible light spectrum (approximately 400 nm to 700 nm). The yellow color in ZnSe will correspond to a transparency range within the yellow region of the visible spectrum (around 570 nm to 600 nm).

Determine the suitable material for a 1550-nm optical communications wavelength light detector

Using the calculated minimum wavelengths for germanium, silicon, gallium-arsenide, and ZnSe, check which material has a transparency range that can absorb the 1550-nm optical communications wavelength. This means that the minimum wavelength for that material must be less than or equal to 1550 nm. Identify the suitable material for the light detector.

Key concepts

Energy-Wavelength Relationship

When studying semiconductors, one of the key principles is understanding the energy-wavelength relationship. This relationship is described by the equation:
\[ E = \frac{hc}{\lambda} \]
where \(E\) denotes energy of a photon, \(h\) is Planck's constant (approximately \(6.626 \times 10^{-34} \text{J} \cdot \text{s}\)), \(c\) is the speed of light (approximately \(3 \times 10^8 \text{m/s}\)), and \(\lambda\) is the wavelength of the photon. This fundamental equation allows us to calculate the minimum wavelength of light (\(\lambda\)) that can be absorbed by a semiconductor based on its band-gap energy, which is the energy needed to elevate an electron from the valence band to the conduction band. The threshold for absorption corresponds to the band-gap of the material: photons with energy below this threshold cannot excite electrons and thus pass through the material, contributing to its transparency.

Optical Transparency Range

The optical transparency range of a material is the spectrum of wavelengths that can pass through the material without being absorbed. To specify this range for semiconductors, we must look at their band-gap energies. The minimum photon energy required to bridge the band-gap corresponds to the longest wavelength that the semiconductor can absorb. Photons with energies less than the band-gap will not be absorbed, which means that the semiconductor will be transparent to light with wavelengths longer than this minimum value. For instance, a semiconductor with a larger band-gap energy will be transparent to more of the visible spectrum, often appearing as colorless, whereas one with a smaller band-gap may absorb visible light and appear colored, as in the case of ZnSe, which absorbs lower-energy (longer wavelength) yellow light, resulting in the yellow color observed experimentally.

Semiconductor Materials

Semiconductor materials such as germanium, silicon, and gallium arsenide, each have unique band-gap energies that define their electronic and optical properties. These materials are essential in electronics and photonics industries for their ability to conduct electricity under certain conditions—a behavior lying between that of insulators and conductors. The band-gap of a semiconductor dictates not just its transparency range but also its utility in various applications—detecting light, emitting light, or converting sunlight into electricity. Understanding their band-gap energies is crucial for their application in devices such as photodetectors, LEDs, and solar cells. In particular, these materials often find use in light detectors for optical communications, with each material being carefully selected based on its ability to detect specific wavelengths of light.

Optical Communications Wavelength

For telecommunications, the optical communications wavelength typically refers to the wavelength of light used in fiber optic cables to transmit information. The standard wavelength for this purpose is around 1550 nm, as it offers minimal loss, allowing for efficient long-distance communication. It's crucial that the semiconductor material used in photodetectors for this purpose has a band-gap corresponding to an energy less than or equal to the photon energy at 1550 nm. This ensures the material is sensitive to this wavelength and can effectively detect and convert the optical signals into electrical ones. Selecting the appropriate semiconductor with the right band-gap for the intended wavelength is vital for the efficiency and reliability of optical communication systems.