A p - n junction possesses a depletion region that is 400 nm wide,...

Width of the depletion region, $d = 400 \text{ nm} = 400 \times 10^{-9} \text{ m}$

Electric field in the depletion region, $E = 5 \times 10^5 \text{ V m}^{-1}$

The potential barrier across the junction is given by $V = E \times d$

$V = (5 \times 10^5) \times (400 \times 10^{-9}) = 0.2 \text{ V}$

The minimum kinetic energy required for an electron to diffuse from the $n$-side to the $p$-side is equal to the energy required to overcome the potential barrier, which is $K = eV$

$K = e(0.2 \text{ V}) = 0.2 \text{ eV}$