In structures with narrow barriers, the electrons will not travel far enough to suffer collisions as they cross the barrier. Under these circumstances, the thermionic emission theory is a more accurate representation of the current transport [47]. The current density is given by
where 
is the effective Richardson constant given by
If one compares the current density predicted by the diffusion theory (34) to that predicted by the thermionic-emission theory (35), one finds that the dependence upon the barrier height and the applied voltage is identical, and that the theories differ only in the pre-exponential factor. Moreover, if one evaluates the ratio of these factors one finds
where 
is the mean-free-path in one dimension.
The processes modeled by diffusion and by thermionic emission are
effectively in series, so that the current density is determined by that
process which predicts the lower current density. On this basis, the
diffusion theory is appropriate for barriers in which 
,
while the thermionic emission theory is appropriate for barriers for which
.
However, if the barrier becomes very narrow, current transport by quantum-mechanical tunneling becomes more prominent. In many semiconductor heterostructures significant tunneling can occur through barriers of several nanometers thickness due to the low effective mass of the carriers. This may be observed in those heterojunctions which naturally form thin barriers, such as heavily-doped isotype junctions, or in thin heterostructure barriers designed to permit tunneling. The evaluation of the tunneling currents in heterostructures is described in detail in Chapter 9 of the present volume.
The ability to make abrupt steps in the band-edge energy using heterostructures is exploited in hot-electron transistors [5]. Electrons passing over such a barrier into a lower-potential region are suddenly accelerated to high kinetic energies, which can be sufficient to carry them across a sufficiently narrow base region.
