Question

If the tunneling current is 8.7 pA when an STM probe is 0.40 nm from a surface and 17.0 pA when the probe is 0.50 nm from the surface, calculate the current on moving the tip in 0.10-nm steps from 0.40 nm to 1.50 nm.

Short answer

The value of current is

$\begin{array}{|ll|}\hline \mathrm{d},\mathrm{nm}& \mathrm{It}\\ 0.4& 8.7\\ 0.5& 17.0\\ 0.6& 33.2\\ 0.7& 64.9\\ 0.8& 126.8\\ 0.9& 247.8\\ 1.0& 484.3\\ 1.1& 946.4\\ 1.2& 1849.2\\ 1.3& 3613.5\\ 1.4& 7060.9\\ 1.5& 13797.2\\ \hline\end{array}$

Step-by-step solution

Concept:

We have to use equation

$$\begin{gathered} I_t=V{e}^{-Cd} \\ \frac{{\left(I_t\right)}_2}{{\left(I_t\right)}_1}=\frac{e^{-C{d}_2}}{e^{-C{d}_1}} \end{gathered}$$

Use the equation below to find current

$\ln {\left(I_t\right)}_x=\ln {\left(I_t\right)}_1+C\left(d_1-d_x\right)$

Description

First find the ration from the given current value

Given tunneling current is 8.7 pAmp when STM probe is 0.4 nm from surface,
and tunneling current is 17.0 pAmp when the probe is 0.50 nm from surface,

$$\begin{gathered} I_t=V{e}^{-Cd} \\ \frac{{\left(I_t\right)}_2}{{\left(I_t\right)}_1}=\frac{e^{-C{d}_2}}{e^{-C{d}_1}} \\ C=\frac{\ln {\left(I_t\right)}_2-\ln {\left(I_t\right)}_1}{d_1-d_2} \\ Substitutevalues \\ C=\frac{\ln (17.0)-\ln (8.7)}{0.40-0.50} \\ C=-6.699 \end{gathered}$$

We can find the tunneling current for a given distance can be calculated by using a known tunneling current and C from the equation below.

$\ln {\left(I_t\right)}_x=\ln {\left(I_t\right)}_1+C\left(d_1-d_x\right)$

$$\begin{gathered} \ln {\left(I_t\right)}_x=\ln (8.7)-6.699(0.40-0.60) \\ \ln {\left(I_t\right)}_x=3.50 \\ {\left(I_t\right)}_x=e^{3.50} \\ {\left(I_t\right)}_x=33.11\mathrm{pA} \end{gathered}$$

Similarly we can find current for all other distance values and create a table.

Table of values

We can create table as below

$\begin{array}{|lll|}\hline \mathrm{d},\mathrm{nm}& \ln \left({\mathrm{I}}_{\mathrm{t}}\right)& {\mathrm{I}}_{\mathrm{t}}\\ 0.4& 2.1633& 8.7\\ 0.5& 2.8332& 17.0\\ 0.6& 3.5031& 33.2\\ 0.7& 4.1730& 64.9\\ 0.8& 4.8429& 126.8\\ 0.9& 5.5128& 247.8\\ 1.0& 6.1827& 484.3\\ 1.1& 6.8526& 946.4\\ 1.2& 7.5225& 1849.2\\ 1.3& 8.1924& 3613.5\\ 1.4& 8.8623& 7060.9\\ 1.5& 9.5322& 13797.2\\ \hline\end{array}$