# Probability of Quantum Tunneling

• Minhty
In summary, the researcher is trying to calculate the thickness of the oxide layer on a metal-oxide-semiconductor device by applying a positive voltage and assuming that the electrons will tunnel through the oxide. However, they are unsure of how to do the math correctly and are seeking advice.
Minhty

## Homework Statement

I am given a metal-oxide-semiconductor device. I apply a positive voltage (30 V) on to the metal. Theoretically, the electrons should tunnel through the oxide. I want to calculate the oxide thickness for only 5% of electrons tunneling through the oxide.

## Homework Equations

I used the equations from here:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/barr.html

where ψ=e-αx
α = √(2m(U-E))/[STRIKE]h[/STRIKE]

Also, I thought the probability of electrons tunneling is:
|ψ|2 = e-2αx

## The Attempt at a Solution

So I thought that E is electron energy and E≈kT, but my professor told me that it isn't true and didn't explain to me what it is, so I don't know what the electron energy is anymore.

Also, I thought U is the applied voltage. I made the voltage into energy by the equation: voltage = energy/charge so 30 V become 30eV

I don't know if I'm doing this right or if I'm putting the wrong numbers in the wrong place. Any advice is appreciated! Thanks!

Unfortunately you’re not using the right equations at the moment. The wavefunction you have is for electrons inside the barrier, so you are calculating the electron density a distance $x$ inside the barrier rather than the tunnelling probability. Moreover, the shape of the potential in your problem is different from that website –*can you see why?

Do you have any notes from lectures or your textbook that look more relevant to this situation?

I'm sorry but I don't have resources that is relevant to the situation.

I can't help with the math/physics other than to say that when measuring gate oxide breakdown characteristics it was always referred to as Fowler-Nordheim tunneling. Real gate oxides would have defects and be worn out by tunneling? so the practical figure for gate oxide thickness is probably higher than the theoretical. I hope that offers some help, part of my motivation for commenting on an area I can rightly profess ignorance in is to see what the answer actually is.

Last edited:

Hello, thank you for sharing your question. I can provide some insight into the concept of quantum tunneling and how it applies to your given scenario.

First, let's define quantum tunneling. It is a phenomenon in quantum mechanics where a particle can pass through a potential barrier even though it does not have enough energy to overcome the barrier. This is possible due to the wave-like nature of particles at the quantum level.

In your scenario, you have a metal-oxide-semiconductor device with a positive voltage applied to the metal. This creates a potential barrier for the electrons in the metal. According to quantum tunneling theory, there is a finite probability that some electrons will tunnel through the oxide barrier and reach the other side.

Now, onto the calculations. The equations you have used are correct, but there are a few things that need to be clarified. The energy of the electrons in the metal is not equal to kT, where k is the Boltzmann constant and T is the temperature. This is because the electrons in the metal have a range of energies, not just one specific energy. In this case, the energy of the electrons can be approximated by the applied voltage, as you have correctly done. However, it is important to note that this is not the exact energy of the electrons, but rather an approximation.

Additionally, the value of U should not be the applied voltage, but rather the height of the potential barrier that the electrons need to tunnel through. This can be calculated using the applied voltage and the work function of the metal. The work function is the minimum amount of energy needed to remove an electron from the metal.

Finally, to calculate the oxide thickness for 5% of electrons to tunnel through, you can use the equation for the probability of tunneling, which you have correctly identified as |ψ|2 = e-2αx. In this case, you will need to solve for x, the thickness of the oxide. This can be done by taking the natural logarithm of both sides and rearranging the equation to solve for x.

In conclusion, the concept of quantum tunneling is a complex and fascinating one, and it is important to have a solid understanding of the principles and equations involved in order to accurately calculate probabilities and make predictions. I hope this response has provided some clarification and guidance for your homework question. Good luck!

## 1. What is probability of quantum tunneling?

The probability of quantum tunneling is a phenomenon in quantum mechanics where particles have a non-zero probability of crossing a potential energy barrier, even though they do not have enough energy to overcome it classically. This probability is affected by various factors such as the size and shape of the barrier, the energy and mass of the particle, and the strength of the potential barrier.

## 2. How is probability of quantum tunneling calculated?

The calculation of the probability of quantum tunneling involves solving the Schrödinger equation, which describes the evolution of quantum systems over time. The wave function of the particle is used to determine the probability of finding the particle at a certain position, taking into account the potential energy barrier and other relevant factors.

## 3. What is the significance of probability of quantum tunneling?

Quantum tunneling is a crucial concept in many areas of modern physics, including semiconductor devices, nuclear fusion, and radioactive decay. It also plays a role in many natural phenomena, such as the nuclear reactions in stars and the decay of atomic nuclei.

## 4. Can probability of quantum tunneling be observed?

While the probability of quantum tunneling cannot be directly observed, its effects can be observed in various experiments. For example, in scanning tunneling microscopy, the tunneling of electrons across a potential barrier can be measured and used to create images of surfaces at the atomic level.

## 5. How does temperature affect probability of quantum tunneling?

Temperature can have a significant impact on the probability of quantum tunneling. As temperature increases, the kinetic energy of particles also increases, making it more likely for them to overcome potential barriers. This can result in a decrease in the probability of quantum tunneling at higher temperatures.

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