In Chapter 4 we looked at how electrons could be trapped in various examples of potential wells and made to behave as though they were only two-dimensional (or less). In this chapter we shall look at free electrons that encounter barriers or other obstacles as they travel. Again, most of the potential profiles will be one-dimensional and we need only solve the Schrödinger equation in this dimension, although the other dimensions enter into the calculation of the current. We shall use the general tool of T-matrices, which can simply be multiplied together to yield the transmission coefficient for an arbitrary sequence of steps and plateaus. Two particular applications are to resonant tunnelling through a double barrier and to an infinite, regularly spaced sequence of barriers, a superlattice. Two barriers show a narrow peak in the transmission when the energy of the incident electron matches that of a resonant or quasi-bound state between the barriers (Section 5.5). This peak broadens into a band in the superlattice, and Section 5.6 shows how band structure and Bloch's theorem emerge for a specific example.

Many low-dimensional structures cannot simply be factorized into one-dimensional problems but have many leads, each with several propagating modes. These will be treated in Section 5.7 and we shall derive one of the famous results of low-dimensional systems, the quantized conductance.