Question

a. What is the probability that an electron will tunnel through a$0.50nm$air gap from a metal to a STM probe if the work function is $4.0eV$?

b. The probe passes over an atom that is$0.050nm$“tall.” By what factor does the tunneling current increase?

c. If a $10\%$current change is reliably detectable, what is the smallest height change the STM can detect?

Short answer

a. The probability that an electron will tunnel through a $0.50nm$air gap from a metal to an STM probe is $3.4\times {10}^5$.

b. the tunneling current increases with a factor of $2.8$.

c. The smallest height change is$0.00463nm$.

Step-by-step solution

Part (a) step 1: Given Information

We need to find the probability that an electron will tunnel through a$0.50nm$ air gap from a metal to a STM probe.

Part (a) step 2: Simplify

First, we need to find the penetration distance of the electrons into the forbidden region.

$\eta =\frac{h}{\sqrt{2m(U_0-E)}}$
but we are given that the work function is $4\text{eV}$which is equivalent to the term $\left(U_0-E\right)$so,

$$\begin{gathered} \eta =\frac{1.05\times {10}^{34}\text{J}\cdot \text{s}}{\sqrt{2\left(9.11\times {10}^{-31}\text{kg}\right)\left(4\text{eV}\times 1.6\times {10}^{-19}\text{J}/\text{eV}\right)}} \\ =9.72\times {10}^{-11}\text{m} \\ =0.0972\text{nm} \end{gathered}$$

Now, the probability that the electron will penetrate the air barrier is given as:

$P_{tunnel}=e^{2w/\eta }$

Here $\left(w\right)$is the width of the barrier. Since:

$P_{tunnel}=\exp \left(-\frac{1\text{nm}}{0.0972\text{nm}}\right)=3.4\times {10}^5$

Part (b) step 1: Given Information

We need to find the factor does the tunneling current increase or not.

Part (b) step 2: Simplify

The tunneling current is proportional to the tunneling probability as:

$I\propto P_{tunnel}$

so the ratio of the current for a barrier with width $\left(0.5\text{nm}\right)$ to the current for a barrier with width $\left[\left(0.5-0.05\right)=0.45\text{nm}\right]$as:

role="math" localid="1650460494965" $\begin{array}{rr}\frac{I_2}{I_1}=\frac{P_{tunnel,2}}{P_{tunnel,1}}& =\frac{\exp \left(-\frac{0.9\text{nm}}{0.0972\text{nm}}\right)}{\exp \left(-\frac{1\text{nm}}{0.0972\text{nm}}\right)}\end{array}$

$\begin{array}{rr}\frac{I_2}{I_1}& =2.8\end{array}$

the tunneling current increases with a factor of$2.8$.

Part (c) step 1: Given Information

We need to find the smallest height change the STM can detect.

Part (c) step 2: Simplify

The height change equals the difference between the initial width and the new width that caused the increase in the current:

$h=w_i-w_f$

and we are looking for the height that would make the current increase by a factor of $\left(10\%\right)or(0.1)$. Therefore:

$$\begin{gathered} \frac{I_2}{I_1}=1.1=\frac{P_{tunnel,2}}{P_{tunnel,1}} \\ =\frac{\exp \left(-\frac{2w_f}{\eta }\right)}{\exp \left(-\frac{2w_i}{\eta }\right)} \\ =\exp \left(2\frac{w_i-w_f}{\eta }\right) \\ =\exp \left(\frac{2h}{\eta }\right) \end{gathered}$$

take to finding the minimum height $h$.

$\begin{array}{c}\ln \left(1.1\right)=\frac{2h}{\eta }\\ h=\frac{\ln \left(1.1\right)\eta }{2}=\frac{\ln \left(1.1\right)\times 9.72\times {10}^{-11}\text{m}}{2}\\ h=4.63\times {10}^{-12}\text{m}=0.00463\text{nm}\end{array}$