Question

In a particular crystal, the highest occupied band is full. The crystal is transparent to light of wavelengths longer than 295nm but opaque at shorter wavelengths. Calculate, in electron-volts, the gap between the highest occupied band and the next higher (empty) band for this material.

Short answer

The gap between the highest occupied band and the next higher band for this material is 4.20eV .

Step-by-step solution

The given data

• The crystal is transparent to the wavelengths of light larger than $\lambda$while opaque to shorter wavelengths.
• Wavelength of the light, $\lambda =295\mathrm{nm}$

Understanding the concept of band gap energy

The valence band is fully occupied whereas the conduction band is not occupied at all. If an electron in the valence band is to absorb a photon, the energy it receives must be sufficient to excite the electron through the band gap. Photons with energies less than the gap width are not absorbed and the semiconductor is transparent to this radiation, whereas the photons with energies greater than the width gap are absorbed and the electrons jump to the conduction band. Thus, using the given wavelength in Planck's relation, we can get the band gap energy.

Formula:

Plank’s relation for energy, $E=\frac{hc}{\lambda },\mathrm{where}hc=1240\mathrm{eV}.\mathrm{nm}$ (i)

Calculation of the value of the energy gap

the width of the band gap is the same as the energy of a photon associated with a wavelength of 295 nm. Thus, using the given data in equation (i), we can get the value of the energy gap as follows:

$$\begin{gathered} E=\frac{1240\mathrm{eV}.\mathrm{nm}}{295\mathrm{nm}} \\ =4.20\mathrm{eV} \end{gathered}$$

Hence, the value of the energy gap is $4.20\mathrm{eV}$.