The early papers by Klein, Sauter and Hund which investigate scattering off a high step potential in the context of the Dirac equation are discussed to derive the 'paradox' first obtained by Klein. The explanation of this effect in terms of electron-positron production is reassessed. It is shown that a potential well or barrier in the Dirac equation can become supercritical and emit positrons or electrons spontaneously if the potential is strong enough.

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It is shown that a Dirac particle of mass m and arbitrarily small momentum will tunnel without reflection through a potential barrier V = U(c)(x) of finite range provided that the potential well V = -U(c)(x) supports a bound state of energy E = -m. This is called a supercritical potential well.

The Klein paradox is reassessed by considering the properties of a finite square well or barrier in the Dirac equation. It is shown that spontaneous positron emission occurs for a well if the potential is strong enough. The vacuum charge and lifetime of the well are estimated. If the well is wide enough, a seemingly constant current is emitted. These phenomena are transient whereas the tunnelling first calculated by Klein is time-independent. Klein tunnelling is a property of relativistic wave equations, not necessarily connected with particle emission. The Coulomb potential is investigated in this context: it is shown that a heavy nucleus of sufficiently large Z will bind positrons. Correspondingly, it is expected that as Z increases the Coulomb barrier will become increasingly transparent to positrons. This is an example of Klein tunnelling.

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It is shown that the amplitude for reflection of a Dirac particle with arbitrarily low momentum incident on a potential of finite range is −1 and hence the transmission coefficient T = 0 in general. If however the potential supports a half-bound state at momentum k = 0 this result does not hold. In the case of an asymmetric potential the transmission coefficient T will be non-zero whilst for a symmetric potential T = 1. Therefore in some circumstances a Dirac particle of arbitrarily small momentum can tunnel without reflection through a potential barrier.