Resonant Tunneling: Here we investigate the intriguing fact that while a single E

Resonant Tunneling: Here we investigate the intriguing fact that while a single E0 barrier always has a reflected wave, introducing a second barrier can give a reflection probability of zero, known as resonant tunnel­ing and important to many modern electronic devices.

Equation (6-15) gives the wave functions outside a single barrier. (We actually won’t need the solutions inside the barrier.) To make room for a second one, we shift this “right barrier” a distance A’ to the right, so that it extends from x = s to x = s + L, which requires that we also shift the wave functions a distance s.

Right barrier:

If we now replace. x by -x, we obtain solutions of the time-independent Schrodinger equation valid for a "left barrier” extending from x = -s -L, to. x = -s.

However, note that the complex conjugate of the time- independent Schrodinger equation is the same equation but with ψ* in place of ψ, because E, U(x), and all other factors are real. In other words, ψ* is as valid a solution as ψ. Thus, for the left barrier we can instead write

Left barrier:

We choose these because the F*-term represents particles on the left of the left barrier and moving right— incident particles. The F-term in the right-barrier solutions represents particles transmitted through the right barrier. Together, they imply perfect transmission, for |F| = |F*|. We can satisfy the Schrodinger equation everywhere for both barriers if we can make the solutions compatible in the region between them, x = -s to x = +s. By equating the two solutions in this region and using the relationship between B and A given in Exercise 30—a bit messy to derive, but nonetheless true for the E

See Exercises 40 and 56 for applications of this result.