Scanning Tunneling Microscope

[lostinyou7]

Homework Statement

A scanning tunneling microscope (STM) can precisely determine the depths of surface features because the current through its tip is very sensitive to differences in the width of the gap between the tip and the sample surface. Assume that in this direction the electron wave function falls off exponentially with a decay length of 0.108 nm - that is with C=9.26 nm-1. Determine the ratio of the current when the STM tip is 0.489 nm above a surface feature to the current when the tip is 0.512 nm above the surface.

Homework Equations

Probability = e^(-2w/n)

The Attempt at a Solution

I assumed neta(n) was constant regardless of how far it was from the surface.
From there, I equated to two probability equations for the first and second distance.

Let P = probability

P1 = e^(-2*0.489/n)
ln P1 = (-2*0.489)/n
n*(ln P1) = (-2*0.489)

P2 = e^(-2*0.512/n)
ln P2 = (-2*0.512)/n
n = (-2*0.489) / (ln P2)

(-2*0.489)(ln P1) / (ln P2) = (-2*0.489)
(0.489)(ln P1) = (0.489)(ln P2)

Now I'm not too sure where to go from here or if my approach is even correct.

Any help would be appreciated, thanks!

 

[anathema]

Your approach is on the right track, but there are a few things that need to be clarified. Firstly, the equation you are using for probability is incorrect. The correct equation is P = e^(-2w/lambda), where lambda is the decay length. Also, neta(n) is not constant, it is a function of distance from the surface and is given by neta(n) = e^(-2w/n).

With this in mind, we can proceed with the solution. Let's call the ratio of the currents I1/I2. We can express this ratio as:

I1/I2 = (neta(n1)/neta(n2))^2

where n1 is the distance of 0.489 nm and n2 is the distance of 0.512 nm.

Substituting in the values for neta(n1) and neta(n2) and using the correct equation for probability, we get:

I1/I2 = (e^(-2*0.489/0.108)/e^(-2*0.512/0.108))^2

Simplifying this, we get:

I1/I2 = (e^-9.056/e^-9.481)^2

= (0.0001/0.00007)^2

= 1.6327

Therefore, the ratio of the currents is approximately 1.6327. I hope this helps. If you have any further questions, please don't hesitate to ask.