Photodiode Basics and Band Gap Energy

A photodiode is a semiconductor device that converts light into an electrical current. When photons of sufficient energy strike the photodiode, they create electron-hole pairs, leading to a photocurrent. The minimum energy required for a photon to generate an electron-hole pair is determined by the band gap energy (\(E_g\)) of the semiconductor material.

For a photodiode to detect light, the energy of the incident photons must be equal to or greater than the material's band gap energy. This concept is crucial for understanding the operational limits of a photodiode, particularly its cutoff wavelength.

Cutoff Wavelength Formula

The cutoff wavelength (\(\lambda_c\)) represents the longest wavelength of light that a photodiode can detect. Photons with wavelengths longer than the cutoff wavelength will not have enough energy to excite electrons across the band gap, and thus, will not generate a significant photocurrent.

The relationship between the band gap energy (\(E_g\)) and the cutoff wavelength (\(\lambda_c\)) is given by the formula derived from the photon energy equation:

\[E_g = \frac{hc}{\lambda_c}\]

Where:

• \(E_g\) is the band gap energy (in Joules or electron-volts).
• \(h\) is Planck's constant (\(6.626 \times 10^{-34}\) J·s or \(4.135 \times 10^{-15}\) eV·s).
• \(c\) is the speed of light in vacuum (\(3 \times 10^8\) m/s).
• \(\lambda_c\) is the cutoff wavelength (in meters).

For convenience, when \(E_g\) is expressed in electron-volts (eV) and \(\lambda_c\) is desired in nanometers (nm), the formula simplifies to:

\[\lambda_c (\text{nm}) = \frac{1240}{E_g (\text{eV})}\]

We will use this simplified formula to calculate the cutoff wavelengths for Silicon and Germanium photodiodes.

Silicon Photodiode Cutoff Wavelength Calculation

Given the band gap energy for a Silicon photodiode, \(E_{g,Si} = 1.1 \text{ eV}\).

Using the simplified formula for cutoff wavelength:

\[\lambda_{c,Si} = \frac{1240}{E_{g,Si}}\]

\[\lambda_{c,Si} = \frac{1240}{1.1 \text{ eV}}\]

\[\lambda_{c,Si} \approx 1127.27 \text{ nm}\]

Therefore, the cutoff wavelength for a Silicon photodiode is approximately 1127.27 nm.

Germanium Photodiode Cutoff Wavelength Calculation

Given the band gap energy for a Germanium photodiode, \(E_{g,Ge} = 0.67 \text{ eV}\).

Using the simplified formula for cutoff wavelength:

\[\lambda_{c,Ge} = \frac{1240}{E_{g,Ge}}\]

\[\lambda_{c,Ge} = \frac{1240}{0.67 \text{ eV}}\]

\[\lambda_{c,Ge} \approx 1850.75 \text{ nm}\]

Therefore, the cutoff wavelength for a Germanium photodiode is approximately 1850.75 nm.

Summary of Cutoff Wavelengths

The calculated cutoff wavelengths for Silicon and Germanium photodiodes are summarized below:

Comparing these results with the given options, the pair 1127.27 nm and 1850.75 nm aligns with our calculations.