LED Emission Wavelength Calculation

An LED emits photons with energy approximately equal to the semiconductor's bandgap energy ($E_g$). The relationship between photon energy ($E$) and wavelength ($\lambda$) is given by the formula: $E = \frac{hc}{\lambda}$ Where $h$ is Planck's constant and $c$ is the speed of light.

For emission, we set the photon energy equal to the bandgap energy: $E_g = \frac{hc}{\lambda}$ Rearranging this gives the emission wavelength: $\lambda = \frac{hc}{E_g}$

Calculating Emission Wavelength

Given values:

• Bandgap energy, $E_g = 1.3\text{ eV}$
• Planck's constant, $h = 6.63 \times 10^{-34}\text{ J s}$
• Speed of light, $c = 3 \times 10^8\text{ m s}^{-1}$

First, convert the bandgap energy from electronvolts (eV) to Joules (J): $1\text{ eV} \approx 1.602 \times 10^{-19}\text{ J}$ $E_g = 1.3\text{ eV} \times (1.602 \times 10^{-19}\text{ J/eV}) \approx 2.0826 \times 10^{-19}\text{ J}$

Now, calculate the central emission wavelength:

Convert the wavelength to nanometers (nm): $1\text{ m} = 10^9\text{ nm}$ $ \lambda \approx 9.55 \times 10^{-7} \times 10^9\text{ nm} \approx 955\text{ nm} $

Identifying Non-Emitting Wavelength Ranges

The LED is expected to emit light primarily around the calculated wavelength of 955 nm. We need to find the options representing wavelength ranges where the LED will NOT emit.

Let's examine each option's range relative to the calculated 955 nm emission wavelength:

• Option A: $1410 \pm 20\text{ nm}$ covers the range [1390 nm, 1430 nm]. This is significantly longer wavelength (lower energy) than 955 nm.
• Option B: $1090 \pm 20\text{ nm}$ covers the range [1070 nm, 1110 nm]. This is longer wavelength (lower energy) than 955 nm.
• Option C: $950 \pm 20\text{ nm}$ covers the range [930 nm, 970 nm]. This range includes the calculated emission wavelength of 955 nm. The LED will emit here.
• Option D: $510 \pm 20\text{ nm}$ covers the range [490 nm, 530 nm]. This is significantly shorter wavelength (higher energy) than 955 nm.

Therefore, the LED will NOT emit in the wavelength ranges corresponding to options A, B, and D, as these are too far from the energy dictated by the bandgap.