Applications of ``tunneling''

The concept of tunneling, whereby a quantum particle penetrates through a classically disallowed region, has also been applied to many situations with spectacular success. Three examples suffice.

1. A new ultra-Microscope : A new class of microscopes (with atomic resolution) that exploit the tunneling current between a specimen and a very sharp tip has been developed. The sequence of three figures 18, 19 and 20 below show the tunneling tip of a scanning tunneling microscope (STM), the scanning principle, and an image at atomic resolution of a silicon crystal surface.

   Figure 18: The tunneling tip in a tunneling microscope.

   

   Figure 19: The tunneling tip is scanned over the specimen, producing an image of the tunneling current.

   

   Figure 20: The tunneling tip is scanned over the specimen, producing an image of the tunneling current.An image of a silicon crystal surface produced by a STM.

   
2. Alpha particle deacy : The explanation of alpha particle decay as the tunneling of an alpha particle out of the nucleus (figure 21) explained the tremendous variation of alpha-particle lifetimes (25 orders of magnitude, figure 22) as being due to the rather small differences in barrier parameters. This remains one of the most impressive ranges of applicability of a single theory in physics.

   Figure 21: The representation of alpha particle decay as the tunneling of an alpha particle out of the nucleus.

   

   Figure 22: The large variation of alpha particle decay rates or inverse lifetimes (25 orders of magnitude) explained by tunneling theory.

   
3. The end of the road for the classical computer : So far, miniturisation and very large scale integration of microchips has proceeded by pushing engineering technology boundaries. How far can we continue this game ? It turns out that the tunneling process represents a physics limit for the miniaturisation of feature size on a chip. No technological process can go beyond this boundary without changing the physical basis of the computational device. From this point on, we are in the realm of the quantum computer, and the clasical computer can go no further. The tunneling could be between neighbouring wires or across the gate of a transistor or some other feature where the quantum behaviour is manifested.

   Figure: Classical and quantum pathways for an electron to escape its boundary.

   
   Imagine that the quantum process (tunneling) should not be more likely than the classical process (thermally driven over-barrier hopping - Arrhenius Law), so that the limiting case is when they are equal (see figure ).
   

   $$\mbox{Transmission probability through a barrier} \qquad T = e^{-2\frac{\sqrt{2m(U-E)}}{\hbar}L}$$

   $$\mbox{and}$$

   $$\mbox{Thermal over-barrier hopping probability} \qquad P = e^{-\frac{(U-E)}{kT}}.$$ (107)

   
   

Now suppose you imagine typical values for the parameters in the formula. The hight of the barrier above the particle is $(U-E)\approx 1$ eV, and the system is at room temperature $kT = 26$ meV.

$$e^{-2\frac{\sqrt{2m(U-E)}}{\hbar}L} = e^{-\frac{(U-E)}{kT}}.$$ (108)

We find the value for the minimum feature size $L$ of a conventional chip is

$$L \approx 4 \quad \mbox{nm}.$$ (109)

(Hint ; use $\hbar c = 197$ MeV.fm (or ev.nm) and $mc^2=m_ec^2 = 511$ keV)