Scanning Tunneling Microscope Question (introductory modern physics)

[Potato21]

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 0.124 nm - that is with C=8.06 nm-1. Determine the ratio of the current when the STM tip is 0.488 nm above a surface feature to the current when the tip is 0.516 nm above the surface.

Homework Equations

I honestly don't have any idea, the question appears foreign to me. If anything that the penetration depth formula n=(h bar)/sqrt(2m(U-E)) but I don't see how to use it. I'm completely lost

The Attempt at a Solution

I don't really understand what its about so I haven't attempted a solution. The textbook we're using (Knight) doesn't cover any questions like it either. Help!

Thank you in advance.

 

[anathema]

Hello,

I understand your confusion with this question, it can seem quite daunting at first. The key to solving this problem is understanding the concept of tunneling current in an STM and how it relates to the distance between the tip and the sample surface.

In an STM, a voltage is applied between the tip and the sample, creating a potential difference. This potential difference allows electrons to tunnel through the gap between the tip and the sample surface. The current through the tip is then measured and used to create an image of the surface features.

The formula you mentioned, n=(h bar)/sqrt(2m(U-E)), is known as the tunneling probability formula. It relates the probability of an electron tunneling through a barrier to the distance between the tip and the sample surface, represented by n. In this case, n is equal to the decay length of 0.124 nm.

To solve this problem, we need to use the ratio of the tunneling probabilities at two different distances, 0.488 nm and 0.516 nm. This can be calculated using the formula: I/I0 = e^(-2nd), where I is the current at a distance d and I0 is the current at a reference distance (in this case, 0.516 nm).

Using this formula, we can calculate the ratio of the current at 0.488 nm (I) to the current at 0.516 nm (I0). This ratio will give us the answer to the question, as it represents the sensitivity of the current to the distance between the tip and the sample surface.

I hope this helps you understand the problem better and gives you a starting point to solve it. If you have any further questions, please don't hesitate to ask.