Animated Solution for Physics - Semiconductors: The electrical conductivity of a semiconductor increases when electro magnetic radiation of wavelength shorter than $2480\text{ nm}$ is incident on it. The band gap (in $\text{eV}$) for the semiconductor is

Imagine a world where materials can decide whether they want to conduct electricity or not. Welcome to the fascinating realm of semiconductors! Unlike metals, which are always ready to let electrons flow, or insulators, which stubbornly block them, semiconductors have a conditional pass. This condition is governed by something called the Band Gap.

In the atomic structure of a semiconductor, electrons reside in the Valence Band. Above this band lies the Conduction Band, which is empty at absolute zero temperature. The gap between these two bands is the forbidden zone, the band gap ($E_g$). For a semiconductor to conduct electricity, electrons must cross this gap. But they can't do it alone; they need a push.

This is where light comes into play. When electromagnetic radiation (light) falls on a semiconductor, it bombards the material with tiny packets of energy called photons. If a photon has enough energy—specifically, energy greater than or equal to the band gap ($E\ge E_g$)—it can knock an electron from the valence band right into the conduction band.

Once in the conduction band, the electron is free to move, and voilà, the electrical conductivity of the semiconductor increases! This phenomenon is the core principle behind solar cells and light detectors.

The problem states that conductivity increases when the wavelength of the incident light is shorter than $2480\text{ nm}$. Why shorter? Because energy and wavelength have an inverse relationship, given by Planck's equation:

A shorter wavelength means higher energy. Therefore, the threshold wavelength, the absolute maximum wavelength that can just barely excite an electron, is $2480\text{ nm}$. At this exact wavelength, the photon's energy perfectly matches the band gap energy ($E_g$).

To find the band gap, we need to calculate the energy of a photon with a wavelength of $2480\text{ nm}$. While we could plug in the values of Planck's constant ($h$) and the speed of light ($c$) in standard SI units, there is a much faster, battle-tested shortcut for JEE and NEET:

First, let's convert our wavelength from nanometers to Angstroms. Since $1\text{ nm}=10\text{ A˚}$, we have:

Now, substitute this into our master equation:

Look at how perfectly those numbers align!

The band gap of the semiconductor is exactly $0.5\text{ eV}$. This means any photon with energy greater than $0.5\text{ eV}$ (which corresponds to a wavelength shorter than $2480\text{ nm}$) will successfully increase the material's conductivity.