Question

If a material has a band gap ($E_g$) of 4.0 eV, it will most likely be

• A. transparent to visible light
• B. opaque and metallic
• C. a p-type semiconductor at room temperature
• D. highly absorbent in the infrared region

Hint:

A simple rule of thumb for optical transparency:
- $E_g > 3.1 \text{ eV}$: Insulator, transparent to visible light.
- $E_g < 1.8 \text{ eV}$: Semiconductor, absorbs visible light (appears colored or opaque).
- $E_g \approx 0 \text{ eV}$: Metal, reflective and opaque.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks to predict the optical behavior of a solid material with a wide electronic band gap energy ($E_g = 4.0 \text{ eV}$).

Step 2: Key Formula or Approach:

• The energy of an optical photon is given by: \[ E = \frac{hc}{\lambda} \] where $h$ is Planck's constant, $c$ is the speed of light, and $\lambda$ is the wavelength.

• The spectrum of visible light spans wavelengths from approximately $400 \text{ nm}$ (violet) to $700 \text{ nm}$ (red).

• Converting these wavelengths into photon energies:
- Red light ($700 \text{ nm}$): $E \approx 1.8 \text{ eV}$
- Violet light ($400 \text{ nm}$): $E \approx 3.1 \text{ eV}$
Therefore, the energy range of visible light photons is $1.8 \text{ eV}$ to $3.1 \text{ eV}$.

Step 3: Detailed Explanation:

• For an electron in the valence band of a material to absorb a photon, the photon's energy must be equal to or greater than the bandgap energy: \[ E_{\text{photon}} \ge E_g \]
• If $E_{\text{photon}} < E_g$, the material cannot absorb the photon, and the light passes through without attenuation (transparent).

• Since the maximum energy of any visible light photon is $3.1 \text{ eV}$, and the material's bandgap is $4.0 \text{ eV}$: \[ E_{\text{visible}} \le 3.1 \text{ eV} < 4.0 \text{ eV} \]
• Visible light photons do not have enough energy to excite electrons across the $4.0 \text{ eV}$ bandgap.

• As a result, the material cannot absorb visible light, making it transparent to the visible spectrum (such as wide-bandgap insulators like quartz or alumina).

Step 4: Final Answer:
The material with a band gap of 4.0 eV will most likely be transparent to visible light.