What is tunneling?

I first learned about quantum tunneling from science fiction, specifically a short story by Larry Niven.  The idea is often tossed out there as one of those "quantum is weird and almost magical!" concepts.  It is surely far from our daily experience.

Imagine a car of mass \(m\) rolling along a road toward a small hill.  Let’s make the car and the road ideal – we’re not going to worry about friction or drag from the air or anything like that.   You know from everyday experience that the car will roll up the hill and slow down.  This ideal car’s total energy is conserved, and it has (conventionally) two pieces, the kinetic energy \(p^2/2m\) (where \(p\) is the momentum; here I’m leaving out the rotational contribution of the tires), and the gravitational potential energy, \(mgz\), where \(g\) is the gravitational acceleration and \(z\) is the height of the center of mass above some reference level.  As the car goes up, so does its potential energy, meaning its kinetic energy has to fall.  When the kinetic energy hits zero, the car stops momentarily before starting to roll backward down the hill.  The spot where the car stops is called a classical turning point.  Without some additional contribution to the energy, you won’t ever find the car on the other side of that hill, because the shaded region is “classically forbidden”.  We’d either have to sacrifice conservation of energy, or the car would have to have negative kinetic energy to exist in the forbidden region.  Since the kinetic piece is proportional to \(p^2\), to have negative kinetic energy would require \(p\) to be imaginary (!).

However, we know that the car is really a quantum object, built out of a huge number (more than \(10^27\)) other quantum objects.  The spatial locations of quantum objects can be described with “wavefunctions”, and you need to know a couple of things about these to get a feel for tunneling.  For the ideal case of a free particle with a definite momentum, the wavefunction really looks like a wave with a wavelength \(h/p\), where \(h\) is Planck’s constant.  Because a wave extends throughout all space, the probability of finding the ideal free particle anywhere is equal, in agreement with the oft-quoted uncertainty principle. 

Here’s the essential piece of physics:  In a classically forbidden region, the wavefunction decays exponentially with distance (mathematically equivalent to the wave having an imaginary wavelength), but it can’t change abruptly.  That means that if you solve the problem of a quantum particle incident on a finite (in energy and spatial size) barrier from one side, there is always some probability that the particle will be found on the far side of the classically forbidden region.  

This means that it’s technically possible for the car to “tunnel” through the hillside and end up on the downslope.  I would not recommend this as a transportation strategy, though, because that’s incredibly unlikely.  The more massive the particle, and the more forbidden the region (that is, the more negative the classical kinetic energy of the particle would have to be in the barrier), the faster the exponential decay of the probability of getting through.  For a 1000 kg car trying to tunnel through a 10 cm high speed bump 1 m long, the probability is around exp(-2.7e20).  That kind of number is why quantum tunneling is not an obvious part of your daily existence.  For something much less massive, like an electron, the tunneling probability from, say, a metal tip to a metal surface decays by around a factor of \(e^2\) for every 0.1 nm of tip-surface distance separation.  It’s that exponential sensitivity to geometry that makes scanning tunneling microscopy possible.

However, quantum tunneling is very much a part of your life.  Protons can tunnel through the repulsion of their positive charges to bind to each other – that’s what powers the sun.  Electrons routinely tunnel in zillions of chemical reactions going on in your body right now, as well as in the photosynthesis process that drives most plant life. 

On a more technological note, tunneling is a key ingredient in the physics of flash memory.  Flash is based on field-effect transistors, and as I described the other day, transistors are switched on or off depending on the voltage applied to a gate electrode.  Flash storage uses transistors with a “floating gate”, a conductive island surrounded by insulating material, some kind of glassy oxide.  Charge can be parked on that gate or removed from it, and depending on the amount of charge there, the underlying transistor channel is either conductive or not.   How does charge get on or off the island?  By a flavor of tunneling called field emission.  The insulator around the floating gate functions as a potential energy barrier for electrons.  If a big electric field is applied via some other electrodes, the barrier’s shape is distorted, allowing electrons to tunnel through it efficiently.  This is a tricky aspect of flash design.  The barrier has to be high/thick enough that charge stuck on the floating gate can stay there a very long time - you wouldn’t want the bits in your SSD or your flash drive losing their status on the timescale of months, right? - but ideally tunable enough that the data can be rewritten quickly, with low error rates, at low voltages.