Question

Electrons are fired at a rectangular potential energy barrier, once every 173 ms. If the barrier is 2.55 nm thick and has a height that exceeds the energy of the incident electrons by exactly 662 meV, how long on average would you expect to wait for one electron to pass through the barrier?

Answer 1

Given that electrons are fired at a rectangular potential energy barrier, once every 173 ms
thickness = 2.55 nm
= 2.55*10^-9 m
ħ = 1.055*10^-34 Js

TRansmission coefficient for the electron,
T = e^-2 sqrt(m(v-E)/ħ^2)L

sqrt(m(v-E)/ħ^2) = sqrt((9.1*10^-31*662*10^-3*1.6*10^-19)/(1.055*10^-34)^2)
= 2.94*10^9

Now,
T = e^-2*(2.94*10^9)(2.94*10^-9)

= e^-17.28

= 1.86*10^-8

Hence tunnelling probability per attempt = 1.86*10^-8

So total of 1/1.86*10^-8 attempts need to be made = 5.37*10^7 electrons need to hit the potential barrier

Time taken = 5.37*10^7*0.173
= 9.3*10^6 s