Probability of Quantum Tunneling

[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!

 

[Modulated]

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?

 

[Minhty]

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

 

[NoPoke]

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.

 

[paranormal]

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!