# Quantum tunneling

Jack
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Greetings !
Originally posted by Jack
What is quantum tunneling?
The square of the Wave Function of a particle
discribes the probability of finding a
particle in certain points in space.
This probability(theoreticly) is never
0 anywhere.

According to Heisenberg's Uncertainty Principle
the mommentum/location, time/energy of a
particle are never known exactly. This also
means that for a sufficiently short time period
a particle can gain a lot of energy provided
it is "returned" afterwards.

Now, if you consider a certain particle
somewhere it is theoreticly possible according
to the HUP that for a short enough time
period it will gain a sufficient amount of
energy to overcome the potential energy barrier
between different points in space and end
up at that other point.

Qunatum tunneling is limmited by the speed
of light above the Plack scale that is the
limmit of possible certainty/uncertainty.
It is usefull in electronics when electrons
can tunnel "over" a potential difference.

Live long and prosper.

Also, all chemical and nuclear reactions (including life as a consequence of chemical reactions) are results of tunneling.

As well as most properties of solids are the result of tunneling (of electrons away from their hosting atoms).

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Jack
So it's not just something that can happen, it actually does happen. . How common is it?

Twist two wires together and hook up a voltage potential across them. The electron has to tunnel through the barrier of the air that exists between the twisted together wires.

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Twist two wires together and hook up a voltage potential across them. The electron has to tunnel through the barrier of the air that exists between the twisted together wires.
Mmm.. no.. sorry. This is not quantum tunneling!

Quantum tunneling happens all the time -- it is the reason the Sun is able to produce energy. Without the tunneling mechanism, much higher temperatures would be required in the Sun's core -- but the Sun would not be massive enough to generate those temperatures.

Normal electrical circuits do not involve tunneling, but there are a variety of specialized devices that do -- for example, the aptly named 'tunnel diode.'

In addition, we use quantum tunneling in electron microscopy. :)

- Warren

All semiconductors and most poor conducting metals conduct current due to tunneling (atom-to-atom jump of electron from time to time).

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Originally posted by Alexander
All semiconductors and most poor conducting metals conduct current due to tunneling (atom-to-atom jump of electron from time to time).
I'm sorry, that's incorrect. The electrons have enough thermal energy to cross the energy barrier. There is no tunneling in normal semiconductors or poor conductors.

- Warren

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Originally posted by chroot
I'm sorry, that's incorrect. The electrons have enough thermal energy to cross the energy barrier. There is no tunneling in normal semiconductors or poor conductors.

chroot I'm impressed
you understand all these cases: electrons in semiconductors, or crossing a gap between wires, protons overcoming resistance to fusion at lower than expected temperature, and so forth.

Could you give me a rule that I can apply so as to be able to distinguish cases of true tunneling?

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Originally posted by marcus
Could you give me a rule that I can apply so as to be able to distinguish cases of true tunneling?
Find the height of the energy barrier. Determine whether or not the interaction is classically possible. If it's not classically possible, but yet still occurs, it's quantum tunneling. :)

- Warren

Copper oxide can easily have an energy barrier of >10 eV. An electron with less than this will not cross the barrier classically.

Originally posted by chroot
Quantum tunneling happens all the time -- it
is the reason the Sun is able to produce
energy. Without the tunneling mechanism,
much higher temperatures would be required
in the Sun's core -- but the Sun would not
be massive enough to generate those temperatures.
Fascinating !
Could you, please, put an approxiamte number
on the scale of the difference made by q.t.
in this case ?
Thanks !

Live long and prosper.

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Copper oxide can easily have an energy barrier of >10 eV. An electron with less than this will not cross the barrier classically.
Perhaps you should do a little more research. I'd love to see some references.

(Psst: there's no quantum tunneling going on here.)

- Warren

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Originally posted by chroot
Perhaps you should do a little more research. I'd love to see some references.

(Psst: there's no quantum tunneling going on here.)

- Warren

Warren I am guessing you hold the high cards here.
In QTM, quantum tunneling microscopy, there is a gap between
the probe and the surface being probed.

Current flows across this gap. I forget easily----what is the size of this gap (1) as distance and (2) as energy barrier?

Then the question is, what size voltage is applied to the probe relative to the sample so that current flows. This voltage will---according to the criterion you gave---be lower than the barrier voltage. Because actual tunneling happens.

I'm sure you know the figures here because I believe microscopy is one area where you are working or at least know a lot. So please fill out the details for me rather than my having to slog thru a websearch. Interesting thread and not too hard.

Originally posted by chroot
I'm sorry, that's incorrect. The electrons have enough thermal energy to cross the energy barrier. There is no tunneling in normal semiconductors or poor conductors.

- Warren

Incorrect. Explain, how 0.025 eV electron (room temperature) can jump even 0.1 eV gap between atoms without tunneling (and usually gaps in poor conductors are even higher, say 0.8 eV in Si).

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Originally posted by Alexander
Incorrect. Explain, how 0.025 eV electron (room temperature) can jump even 0.1 eV gap between atoms without tunneling (and usually gaps in poor conductors are even higher, say 0.8 eV in Si).
Since when did ALL the electrons have only 0.025 eV of energy? I thought we were talking about THERMAL energy, which is a DISTRIBUTION.

Futhermore, the gaps you speak of in semiconductors are between the valence and conduction bands -- and guess what? The electrons are generally considered to be free-moving once they have been promoted to the conduction band.

The electrons are able to get into the conduction band solely due to their thermal energy. The distribution of occupation of quantum states is given by the Fermi-Dirac integral, and approximated by the Boltzmann distribution. There is no "quantum tunneling" between the valence and conduction bands, I'm sorry.

The "Fermi energy" is the level of the highest state occupied by any electron when the semiconductor is at absolute zero. At any temperature above absolute zero, some electrons are naturally in states higher than the Fermi energy -- all due to our good ol' friend translational kinetic energy -- or heat.

I am using as a reference the book "Semiconductor Physics & Devices" by Neamen. Perhaps you should give me your references?

- Warren

Alexander doesn't need a reference, guaranteed.

The time independent schroedinger equation is
d2phi /dx2= ((2m(U-E))/hbar2) phi (x)

Since d2/dx2e(+-alpha x)= alpha2 e(+- alpha x)
,

alpha = (2m(U-E))1/2/hbar
where m = mass of particle, U = energy barrier, E= particle energy
penetration depth = 1/ alpha

So when U > E, tunneling must happen. the Transmission coefficient that can quantify the energy of the particles that tunneled is dependant upon the distance between the materials as well.

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The time independent schroedinger equation is
Uh.. yeah, thanks -- I understand tunneling... however, it doesn't happen in the situations listed here. :)

- Warren

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Originally posted by chroot
Uh.. yeah, thanks -- I understand tunneling... however, it doesn't happen in the situations listed here. :)

- Warren

I mentioned the Quantum Tunneling Microscope (QTM) in my last post. Does it happen in that situation?

One would assume it does due to the name.

And if so how does your criterion apply to prove it does.
Presumably there is a small gap between the probe and the sample and the energy barrier represented by this gap can be
estimated back-of-envelope style in a couple of minutes or less
by someone familiar with today's microscopes. I apologize if
I am mistaken---ignore this request if it is any real difficulty.

Also I liked your example of the core of the sun. I happen to know the temp is 15 million kelvin which is 1300 eV. So protons are bumping each other with energy of several thousand eevee and you are telling me the barrier to their sticking to each other is much higher than that. What is the number that the proton-proton fusion chain has to tunnel thru---since you say that their thermal 1300 eV is not enough?

Would very much like some more concrete examples.

Uh.. yeah, thanks -- I understand tunneling... however, it doesn't happen in the situations listed here. :)
You do? then you understand that U > E, tunneling happens. Feynman used the twisted wire situation as an example of tunneling.

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You do? then you understand that U > E, tunneling happens. Feynman used the twisted wire situation as an example of tunneling.
I hope you understand that thermal energies of electrons in a wire can be much higher than several tenths of an electron volt.

And can you provide a reference for Feynman's comment?

- Warren

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usually one does not worry about misspellings

Feynman used the twisted wire situation as an example of tunneling.

Could it be that as you make the voltage lower and lower there is a gradual increase in the importance of tunneling?

so twist wires together and apply a very small voltage and
cool the junction down to where most of the electrons don't have much thermal energy and then maybe the person who says "no tunneling" has to reconsider?

Anyway, I do not have an opinion about this but I am bothered by the misspelling of Schwarzschild. Usually I enjoy misspellings on bulletin boards just like everybody else, and indulge in them freely. But I am troubled by this one.

He died that same year (1916) that he figured out the black hole radius and modeled the black hole. Not yet 33 years old. He was a soldier in the WWI and at some front or other and caught an illness and died in May 1916. Yet General Relativity was first published in 1916! How did Karl do this while fighting in a horrible war and getting sick yet almost immediately after Einstein published there he was with
the radius of the event horizon 2GM/c2.

His name means black forehead because, I guess, black hair was not so common so a bunch of it falling over the forehead of one of his ancestors was noticed as a distinguishing characteristic.

When ever I see your posts, which I appreciate and enjoy on occasion as well, I think "scwarz child" instead of "schwarz schild", that is of a child rather than a forehead, and I inevitably have an impulse to ask you to change it out of respect for this person

Originally posted by chroot
Mmm.. no.. sorry. This is not quantum tunneling!

Umm... sorry... yeah... it is. If an electron jumps the vacuum potential, that's quantum tunneling.

Normal electrical circuits do not involve tunneling, but there are a variety of specialized devices that do -- for example, the aptly named 'tunnel diode.'

Uhh... what the hell are you talking about? What do you think resistance is? Understanding electron-impurity scattering requires a little something called the Schroedinger equation... and electron-negative-impurity transmission involves a little condition called E < U.

eNtRopY

P.S. Despite the name, the most prominent mechanism in a zener diode, or 'tunnel diode' as you like to call it, is actually avalanching not tunneling.

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It was in 6 Easy Pieces I believe.

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Originally posted by eNtRopY
P.S. Despite the name, the most prominent mechanism in a zener diode, or 'tunnel diode' as you like to call it, is actually avalanching not tunneling.
A zener diode is not the same thing as a tunnel diode.

- Warren

Originally posted by marcus

...Also I liked your example of the core of the sun. I happen to know the temp is 15 million kelvin which is 1300 eV. So protons are bumping each other with energy of several thousand eevee and you are telling me the barrier to their sticking to each other is much higher than that. What is the number that the proton-proton fusion chain has to tunnel thru---since you say that their thermal 1300 eV is not enough?

Would very much like some more concrete examples.

Coulomb potential at ~1fm from proton (distance at which nuke forces start pulling protons together) is V~kq/r~1.4 MV,so 1.3 KeV is indeed much smaller than about 1.4MeV required for instant fusion. That is why Sun does not desintegrate immediately like supernova. Usually it takes for each proton about 10-15 billion years of non-stop bumping into its neighbors in Sun's interior till a proton "leaks" via such tall (and fat) barrier and fuses with one other proton.

So fusion in Sun's core is soooooo slow that just an ordinary candela releases about million times more energy per cm3 per second than Sun's core does (1 watt/cm3 for a candela vs 1 microwatt/cm3 for Sun's core).

If we reproduce Sun's core conditions in lab, we get totally useless source of energy. Ordinary flame is about million times more powerful.

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Originally posted by Jack
What is quantum tunneling?

Perhaps the simplest possible way to see it is through uncertainty relation (this is really what is happening here - nothing complicating)

Imagine you trapped a particle inside some potential barrier - say some bonding electromagnetic potential that traps particle or, some nuclear potential that traps alpha particle etc. In quantum mechanics you will always have an uncertainty between momentum and position. So, you will have a probability to catch a particle even outside of barrier. That is called tunneling. Examples are everywhere. As posts above already mentioned many semi-conductor devices use it. Also, radio active nuclei radiate through quantum tunneling (indeed you can compute decay rate using quantum mechanics.)

Instanton