How Does Fowler-Nordheim Tunneling Apply to Scanning Tunneling Microscopes?

[Janerio]

Homework Statement

For a project I am doing I have to use the concept of Fowler-Nordhiem tunneling to model a Scanning Tunneling Microscope. Basically we have to relate the current generated from the tip to the field to the tunneling probability. Then relate that current to the applied voltage and height of the tip from the sample. The project can be solved analytically or through approximating solutions using MATLAB (majority of the students usually use MATLAB) The prof suggested that we use a right angled triangular potential barrier and gave us the following equation for the tunneling. The use of this equation is however optional.

Homework Equations

[itex]\inte^{-(2*root(2))/hbar}(V(x) - E)^1/2dx[/itex]

The Attempt at a Solution

I decided the best way to do this question is to find and graph the transmission probability with respect to the potential barrier and the distance from between the tip and sample in the microscope (using planar waves). However, the concept of Fowler-Nordhiem tunneling is a little confusing to me because I don't know how using a triangular potential barrier will change anything or what the significance of it is. Is it supposed to resemble the sharp tip of the microscope? I think the analytical soln should be similar to that in this website


http://ecee.colorado.edu/~bart/book/msfield.htm

but to what level does the WKB (Wigner, Kramers, Brillouin) approximation hold? Please help me understand the significance of the triangular potential barrier and the WKB approximation.

Thanks

 

[mark726]

for posting your question on the forum! I can definitely help clarify some of the concepts you are struggling with. First of all, the concept of Fowler-Nordhiem tunneling is an important phenomenon in quantum mechanics that describes the ability of particles to cross a potential barrier that they would not have enough energy to overcome in classical mechanics. In the context of your project, it is used to explain how electrons can tunnel from the tip of the microscope to the sample, allowing for imaging at the atomic scale.

Now, onto the triangular potential barrier. This is a common model used in quantum mechanics to represent a sharp, high potential barrier. In the case of your project, it can be seen as representing the sharp tip of the scanning tunneling microscope. Using this model allows for a simpler and more intuitive understanding of the tunneling process.

As for the WKB approximation, it is an approximation method used to solve the Schrodinger equation for a potential barrier that is not too steep. In this case, the triangular potential barrier is a good candidate for using the WKB approximation. However, it is important to note that this is still an approximation and may not hold for all scenarios. It would be best to check with your professor or consult other sources to determine the validity of the WKB approximation for your specific project.

I hope this helps to clarify some of the concepts and their significance in your project. Good luck with your project!