If an arbitrary potential barrier is segmented into N regions with constant
potentials (see Fig. 3.9) the wave function in each region can be written as
the sum of an incident and a reflected wave [93]
with the wave number
. The wave amplitudes , , the
carrier mass , and the potential energy are assumed constant for
each region . With the interface conditions for energy and momentum
conservation

(3.73) | |||

(3.74) |

the outgoing wave of a layer relates to the incident wave by a complex transfer matrix:

The transfer matrices are of the form

(3.76) |

with the phase factor . The transmitted wave in Region N can then be calculated from the incident wave by subsequent multiplication of transfer matrices:

If it is assumed that there is no reflected wave in Region N and the amplitude of the incident wave is unity, (3.77) simplifies to

and the transmission coefficient can be calculated from (3.54). The transfer-matrix method based on constant potential segments has the obvious shortcoming that, for practical barriers, the accuracy of the resulting matrix strongly depends on the chosen resolution. A more rigorous approach is to use linear potential segments.