# Maximum distance of quantum tunneling using an STM

## Main Question or Discussion Point

Hey there,

I'm pursuing a degree in Msc. Nanoscale science, and I've been studying about Scanning tunneling microsopes that use Quantum tunneling to study the surface struture of materials.

My question is: What is the maximum distance between the surface that's been studied, and the STM tip for which tunneling can occur?

I understand that the surface has to be quite close, of the order of angstroms or nanometers for the tunneling effects to start taking place, but if I apply a high enough voltage, can I achieve tunneling for a tip that is around a few centimeters to meters away?

What exactly are the parameters that determine tunneling? Is it just the distance? is the surface size of the tip? The voltage? A combination of all three? What are the maximum allowed values for these parameters? Are there any?

As far as I understand it, tunneling is just when electrons tunnel through a barrier, which is in the case of an STM, vaccuum. So, if the voltage is high enough, can the electrons tunnel in? I understand that the probability of tunneling undergoes exponential decay with distance. However, like I stated, if I had a high enough voltage, and if that's the case, how high should my voltage be?

Last edited:

Related Quantum Physics News on Phys.org
In theoretically, there is no exact limitation for the distance.

But In realitically, farther the distance, much more ambiguous the image you derived.

That's because the tunneling current has an exponential relation with the distance, i.e. about exp(-d^0.5). If the distance is set to be far, δI becomes much small.

And the MUCH HIGHER voltage is no helpful.

There's two reason:

1. Additional electrical noise will be introduced.

2. New physical phenomenon will be introduced, i.e. electron emission in a strong electric field.

Technically, there is no limit to the distance and object can tunnel, but the probability gets very unlikely as the distance increases. A better question to ask is how far away must the STM tip be for tunneling to have a reasonable chance of occurring, but then you have to define "reasonable"

I apologize I can't actually find a formula for calculating tunneling transmission probabilities, I have the formula both in my textbook and my notebook from a class I took this quarter, but I left both at my house back at college, don't remember them off the top of my head.

The voltage factors in because it gives the electrons energy, the more energy the electrons have, the greater the tunneling probability, I wouldn't know the effect that STM tip area would have on the results, if any.

Ok, I think I found the formula on the class website, it looks like this:

T being the transmission probability for tunneling, E is the kinetic energy of the particles and U0 being the potential energy for the barrier. L is the length of the barrier, so the distance between the STM tip and the surface you are measuring.

I've not been able to find this formula online, instead I found a much more complicated equation involving integrals, this might be a bit of a simplification, I'm not sure. I think it might assume that the barrier is "rectangular" in a sense, which might not realistic, in nature.

Hey! Thanks for the replies, you two.

But the thing is, the image resolution is not so important for me here. The purpose for which I aim to use the STM in this instance, doesn't demand that. All I need is a tunneling current from the tip of my STM and a metallic conductor that is seperated by vacuum.

Even if the probability is low, and therefore, the current is low, it's okay. I just need some detectable current that I can measure reliably.

However, the distance is very important to me. I need to do it over a few centimeters at least. So the tunneling should occur over a few centimeters. So if I have a good voltage going, then would it occur?

Another question that pops into my mind is if the voltage is that high, would the tip of the STM probe break? I certainly don't want that.

At the same time, I am reminded of experiments where they use a really high voltage to get some electrons into diamond in order to analyze the surface.

I don't need a high current. I don't even need a high voltage. So... just wondering if I could get a reliable current running while using a Tungsten tip a few centimeters away.

Ok, that's a reasonable question. It is pretty tough to answer because some of the values to the equation I posted are rather unknown, or ambiguous.

1) E is related to the voltage, I'm not sure what kind of voltage is in your capacity to run on an STM, I'm not familiar with the technicalities involved with actually operating one, only the theory behind it.

2) I have no clue what kind of potential energy would be gained by crossing a vacuum gap, so U0 is pretty difficult for me to determine with any accuracy.

I'm guessing you want a current of electrons, so m=9.11x10-31kg. Let's set L, distance between the STM tip to the surface at 3cm, 0.03m. Let's assume the barrier height (effectively U0) is twice that of the energy of the electrons. Classically, insurmountable, but according to quantum mechanics, tunneling can occur, so let's see the probability of an electron tunneling through this gap.

L = 0.03m
m = 9.11x10-31kg
U0 = 2E
ħ = 1.05x10-34

Plugging this into the formula, you basically get an answer that looks like T~1/(e^10^9) The denominator is gigantic, so transmission probability is basically zero. You would not be able to generate a reliable current with a STM a few centimeters from the surface. Looking at the equation, the voltage plays a huge role, if you can raise the electron energy to a level very close to the potential barrier height, you might be able to get a steady current.

If we try it again for U0 = 1.01E, still classically insurmountable, but only just. The tunneling probability becomes closer to T~1/(e^10^8), still essentially zero probability, you'd be waiting for years to get a single electron, forget about a current.

So my thought is that you would not be able to achieve any sort of reliable current with the described set up. Even with a voltage very close to the potential barrier height, at a distance in the realm of centimeters, the likelihood of tunneling is still very very tiny. U0/E needs to be around 1.00000001 to start picking up a current, I would think.

Okay. That's brilliant. Thanks a lot soothsayer. You've been a great help.

Kudos to you. :)

Don't mention it, glad I could be of some help!