Band gap and Absorption of Silicon

The band gap of a material refers to the energy difference between the valence band (the highest energy level occupied by electrons) and the conduction band (the lowest energy level that electrons can occupy to become free and conduct electricity). The band gap determines the range of wavelengths of light that a material can absorb.

In the case of silicon, it has a band gap of 1.21 electron volts (eV). This means that silicon can absorb photons with energy equal to or greater than 1.21 eV. To find the corresponding wavelength of light that can be absorbed, we can use the equation:

E = hc/λ

where E is the energy of a photon, h is Planck's constant (6.626 x 10^-34 Js), c is the speed of light (3 x 10^8 m/s), and λ is the wavelength of light.

Calculating the longest wavelength absorbed by silicon

To find the longest wavelength that can be absorbed by silicon, we need to consider the minimum energy required to excite an electron across the band gap. This occurs when the energy of a photon is equal to the band gap energy:

E = 1.21 eV = (1.21 x 1.6 x 10^-19 J)

Substituting this value into the equation and solving for λ:

(1.21 x 1.6 x 10^-19 J) = (6.626 x 10^-34 Js)(3 x 10^8 m/s)/λ

Simplifying the equation gives:

λ = (6.626 x 10^-34 Js)(3 x 10^8 m/s)/ (1.21 x 1.6 x 10^-19 J)

Calculating this gives λ = 1.1 micrometers.

Calculating the band gap of another material

Now, let's consider another material with a longest absorbed wavelength of 0.87 mm (870 micrometers). We can use the same equation to calculate its band gap energy:

E = (6.626 x 10^-34 Js)(3 x 10^8 m/s)/ (0.87 x 10^-3 m)

Simplifying the equation gives:

E = (6.626 x 10^-34 Js)(3 x 10^8 m/s)/ (0.87 x 10^-3 m)

Calculating this gives E = 2.25 x 10^-19 J.

Finally, we can convert the band gap energy into electron volts:

E = (2.25 x 10^-19 J) / (1.6 x 10^-19 J/eV)

This results in E = 1.41 eV.

Therefore, the band gap of the other material is approximately 1.41 eV.