Question

What is the probability that an electron will tunnel through a $0.45\mathrm{nm}$gap from a metal to a STM probe if the work function is $4.0\mathrm{eV}$?

Short answer

The probability is$9.5\times {10}^{-5}.$

Step-by-step solution

Given information 

We have given,

Work function =$4\mathrm{eV}$

penetration depth =$0.45\mathrm{nm}$

We have to find the probability of tunneling.

Simplify 

We know the penetration depth can be written as ,

$\mathrm{\eta }=\frac{\mathrm{h}}{2\mathrm{\pi }\sqrt{2\mathrm{m}({\mathrm{U}}_0-\mathrm{E})}}$

substitute the all value

$$\begin{gathered} \mathrm{\eta }=\frac{(6.634\times {10}^{-34}\mathrm{J}.\mathrm{s})}{2\mathrm{\pi }\sqrt{2\times 9.1\times {10}^{-31}\mathrm{kg}(4\mathrm{eV}\times 1.6\times {10}^{-19})}} \\ \mathrm{\eta }=\frac{(6.634\times {10}^{-34}\mathrm{J}.\mathrm{s})}{2\mathrm{\pi }\times 10.79\times {10}^{-25}} \\ \mathrm{\eta }=0.0972\mathrm{nm} \end{gathered}$$

and we know the probability

$$\begin{gathered} \mathrm{P}={\mathrm{e}}^{-2\mathrm{w}/\mathrm{\eta }} \\ \mathrm{P}={\mathrm{e}}^{-2(0.45\mathrm{nm})/(\mathrm{\eta }=0.0972\mathrm{nm})} \\ \mathrm{P}={\mathrm{e}}^{-0.08748} \\ \mathrm{P}=9.5\times {10}^{-5} \end{gathered}$$