### 3.5.3.1 Piecewise-Constant Potential

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:

 (3.75)

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:

 (3.77)

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

 (3.78)

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.

A. Gehring: Simulation of Tunneling in Semiconductor Devices