Question #43608

A specialized photocathode of thickness $d = 1.5 \text{ \mu m}$ is illuminated by monochromatic light of photon energy $E = 6.0 \text{ eV}$. The illuminated surface is at $x = 0$. Due to a graded alloy composition, the local binding energy (minimum photon energy required to free an electron and leave it at rest at depth $x$) is given by $\Phi(x) = \Phi_0 - \alpha x$, where $\Phi_0 = 4.0 \text{ eV}$ and $\alpha = 2.0 \text{ eV}\cdot\text{\mu m}^{-1}$. Additionally, a non-uniform space-charge distribution creates a local electrostatic potential $V(x) = \frac{\beta}{e} x^2$ inside the material relative to the surface ($V(0) = 0$), where $\beta = 1.0 \text{ eV}\cdot\text{\mu m}^{-2}$ and $e$ is the elementary charge. Assuming electrons travel to the surface without any energy loss other than that due to the macroscopic electrostatic potential, what is the magnitude of the stopping potential required to completely halt the photoelectric current?