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.