Apply a voltage from the tip to the sample. Not enough to drive electrons over.
There's still a barrier for electron flow.
Now add a potential (it can be tiny, say 10 mV) :
Just enough to make the tunneling probability noticeable
=> you start to leak some current, which is measurable.
As L changes, the probability changes a LOT, so the current flow is very sensitive to really tiny changes in L. So, you can "see" effects of a change of L as small as 0.1 A. Now, if you move (scan) the probe tip, you can map the surface height as a function of position. In practice, you change the probe height so the current stays the same. Of course, the current also cares about the local potential on the surface, which is itself determined in part by the electron density. So you are not exactly just measuring the "height" of the surface. But this is clearly a wonderful tool for close in looks at the structure of materials on a sub Angstrom scale.
Lots of solid state devices rely on tunneling. E.g., some diodes, and Josephson junctions... One really beautiful example comes from George Gamow (of the tower!) to explain radioactive alpha decay. It was observed that many nuclei alpha decay, emitting (few MeV) alpha particles, with lifetimes ranging from a microsec to 10 billion years. Gamow modeled the nucleus as a potential well, holding alphas.
Inside, it's an attractive box. Outside, you see the repulsive Coulomb potential caused simply by the net positive charge of the nucleus. (V=2Ze^2/r). At infinity, the potential goes to 0.
A slight change in E makes a big change in both L and alpha <=> (E-V) So, the tunneling probability changes "exponentially". This means that one can have huge variations in lifetime from very small changes in E. Gamow's estimates of lifetime versus energy were in excellent agreement with the data! F+T work out the details, taking into account the fact that the barrier isn't quite square, like we assumed.
You get some sort of "effective L", which you can work out.
The result is
The resulting fit line for the decay probability vs Sqrt[E] has no adjustable parameters.
It's a real prediction, and it works over a range of >24 orders of magnitude of lifetime!! It's a lovely demonstration of wave mechanics in the nucleus.
The essential features for this calculation are just like we had earlier for a square well. The main difference is that
,
rather than
,
which is what we had before.
Rather than doing this integral for the alpha particle case (above), let's look back at the case of the metal at high potential,
(where I use "F" for the E-field strength, so as not to get confused with the
energy E), and this integral then gives
Now, if W=V0-E (the work function), and L=point where V0-eFx=E (i.e., the right side of the barrier), then L=(V0-E)/(eF) = W/eF,
Put in numbers: W=4 eV, and use
1 ev/A = 1.6*10^-19J/(10^-10m)=1.6*10^-9 J/m,
Unless the Electric Field (F) is around a few billion or so V/m, this
exponential kills us. That's a BIG E field! You need very small metal tips, at
a few 100 V, to get something like that. Also, even given an E field as big as
10^9 V/m, then L=W/(eF) =
i.e. "field emission" of electrons is unlikely if L> about 40 A. So, once again, the experiment can measure the current (proportional to T) vs electric field. You expect
Again, many orders of magnitude fit, very successfully.
Mueller first looked at fine details of a metal tip back in 1937 with this tunneling effect, and it's now the basis of many various sorts of "microscopes".
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