Problem 30 - University Physics with Modern Physics

Question

(a) Suppose a piece of very pure germanium is to be used as a light detector by observing, through the absorption of photons, the increase in conductivity resulting from generation of electron-hole pairs. If each pair requires 0.67 \(\mathrm{eV}\) of energy, what is the maximum wavelength that can be detected? In what portion of the spectrum does it lie? (b) What are the answers to part (a) if the material is silicon, with an energy requirement of 1.14 eV per pair, corresponding to the gap between valence and conduction bands in that element?

Step-by-Step Solution

Answer

(a) Germanium detects up to 1850 nm (infrared). (b) Silicon detects up to 1090 nm (infrared).

Step 1: Understand the Energy-Wavelength Relationship

The energy of a photon can be related to its wavelength using the equation: \[E = \frac{hc}{\lambda}\] where \(E\) is the energy in electron volts (eV), \(h\) is Planck's constant \(4.135667696 \times 10^{-15} \, \text{eV}\cdot\text{s}\), \(c\) is the speed of light \(3.00 \times 10^8 \, \text{m/s}\), and \(\lambda\) is the wavelength in meters.

Step 2: Calculate the Maximum Wavelength for Germanium

For germanium, the required energy to generate an electron-hole pair is 0.67 eV. Rearrange the energy-wavelength equation to solve for the maximum wavelength: \[\lambda = \frac{hc}{E}\]. Substitute the values: \(\lambda = \frac{4.135667696 \times 10^{-15} \times 3.00 \times 10^8}{0.67}\). Calculating gives \(\lambda \approx 1.85 \times 10^{-6} \, \text{m}\) or 1850 nm.

Step 3: Identify the Spectrum Portion for Germanium

The wavelength of 1850 nm falls in the infrared portion of the electromagnetic spectrum.

Step 4: Calculate the Maximum Wavelength for Silicon

For silicon, the energy required is 1.14 eV. Using the same formula: \[\lambda = \frac{4.135667696 \times 10^{-15} \times 3.00 \times 10^8}{1.14}\]. Calculating gives \(\lambda \approx 1.09 \times 10^{-6} \, \text{m}\) or 1090 nm.

Step 5: Identify the Spectrum Portion for Silicon

The wavelength of 1090 nm also falls in the infrared portion of the electromagnetic spectrum.

Key Concepts

Germanium light detector

A germanium light detector is a device used to sense light by absorbing photons and increasing its conductivity. This increase occurs due to the generation of electron-hole pairs when the material absorbs light. Germanium is favored because it is sensitive to a broad range of wavelengths, particularly those in the infrared spectrum.

When a germanium-based light detector is exposed to light, each photon absorbed induces energy that creates an electron-hole pair. This enhances the material's ability to conduct electricity. These electron-hole pairs need a specific amount of energy to form. For germanium detectors, this energy is 0.67 electron volts (eV). Thus, only photons carrying this energy or more can create pairs and thus be detected by the germanium detector.

The unique properties of germanium make it particularly effective in applications that require detection of infrared light, such as in night vision technology.

Electron-hole pair energy

The concept of electron-hole pair energy is crucial in understanding how semiconductors like germanium and silicon function as light detectors. An electron-hole pair refers to an electron that is excited and jumps to a higher energy level, leaving behind a 'hole' in the previous energy state.

The energy required to generate an electron-hole pair depends on the material. Germanium needs 0.67 eV, while silicon requires 1.14 eV. This difference represents the energy gap between the valence band and the conduction band in these materials. The smaller the energy gap, the longer the wavelength of light that can be absorbed.

Understanding this principle helps in designing detectors for different wavelength ranges. For germanium, with the required 0.67 eV, the longest detectable wavelength is around 1850 nm, placing it well within the infrared range.

Infrared spectrum detection

Infrared spectrum detection is about sensing light in the infrared region of the electromagnetic spectrum. Light in this spectrum has longer wavelengths than visible light, ranging from about 700 nanometers (nm) to 1 millimeter (mm).

Germanium light detectors are particularly adept at detecting infrared light. When germanium is used, it can detect up to 1850 nm, making it ideal for infrared applications. This capability is vital in many fields including communications, astronomy, and thermal imaging.

The ability to detect infrared light is crucial for various applications where visible light cannot be used due to its short wavelength or intense brightness. Germanium-based detectors thus play an essential role in technologies that need to work in low-light or variable-light conditions.