A p-n junction photodiode is fabricated from a semiconductor with a...

For a photodiode to detect a signal, the energy of the incident photon must be greater than or equal to the band gap energy of the semiconductor.

$E \ge E_g$

$\dfrac{hc}{\lambda} \ge E_g$

$\lambda \le \dfrac{hc}{E_g}$

Substituting the given values:

$\lambda \le \dfrac{6.6 \times 10^{-34} \times 3 \times 10^8}{4 \times 10^{-19}}$

$\lambda \le \dfrac{19.8 \times 10^{-26}}{4 \times 10^{-19}}$

$\lambda \le 4.95 \times 10^{-7} \text{ m}$

$\lambda \le 4950 \text{ \AA}$

Among the given options, only $4000 \text{ \AA}$ is less than $4950 \text{ \AA}$.

Answer: $4000$ Å