Animated Solution for Physics - Semiconductors: LED is constructed from GaAsP semiconducting material. The energy gap of this LED is $1.9\text{ eV}$. Calculate the wavelength of light emitted and its colour. [$h=6.63\times 10^{-34}\text{ Js}$ and $c=3\times 10^8{\text{ ms}}^{-1}$]

Imagine you are looking at the microscopic world inside a Light Emitting Diode (LED). At its core, an LED is a p-n junction diode made from a semiconductor material—in this case, Gallium Arsenide Phosphide (GaAsP).

In any semiconductor, there are two primary energy bands where electrons can exist: the valence band (where electrons are normally bound to atoms) and the conduction band (where electrons are free to move and conduct electricity). The energy difference between these two bands is called the band gap energy ($E_g$). For our GaAsP LED, this gap is given as $1.9\text{ eV}$.

When the LED is forward-biased, electrons from the conduction band drop down to recombine with holes in the valence band. Because the conduction band is at a higher energy level, this drop forces the electron to release its excess energy. In a direct band gap semiconductor like GaAsP, this energy is released as a brilliant flash of light—a photon!

The energy of the emitted photon is exactly equal to the band gap energy of the semiconductor. We can relate the energy of a photon to its wavelength using Planck's famous equation:

Where: - $E$ is the energy of the photon. - $h$ is Planck's constant ($6.63\times 10^{-34}\text{ Js}$). - $c$ is the speed of light in a vacuum ($3\times 10^8\text{ m/s}$). - $\lambda$ is the wavelength of the emitted light.

Since we want to find the wavelength, we can rearrange this equation:

Here is where many students make a silly mistake! The energy $E$ is given in electron-volts (eV), but Planck's constant $h$ is in standard SI units (Joule-seconds). We must convert the energy into Joules before substituting it into our formula.

To convert eV to Joules, we multiply by the charge of an electron ($1.6\times 10^{-19}\text{ C}$):

Now, let's substitute all our values into the rearranged equation:

Simplifying the numerator:

Dividing the terms gives us the wavelength in meters:

To make this number more intuitive, we convert it to nanometers (nm) by multiplying by $10^9$:

We have successfully calculated the wavelength, but what color does $654\text{ nm}$ correspond to?

Recall the visible light spectrum, which spans from approximately $400\text{ nm}$ to $700\text{ nm}$. The lower end ($400\text{ nm}$) corresponds to high-energy violet light, while the upper end ($700\text{ nm}$) corresponds to low-energy red light.

Since $654\text{ nm}$ is situated near the $700\text{ nm}$ mark, the light emitted by this GaAsP LED will be a vibrant red color. Therefore, the correct option is $654\text{ nm}$ and red colour.