The 1D time independent Schrödinger equation can be easily solved analytically in segments of constant potential energy through the matching of the wavefunction and its derivative at every interface at which there is a potential change. The previous lectures showed the process for a single step potential change and a single potential barrier which consts of two interfaces. The process can be generalized for an arbitrary number of interfaces with the transfer matrix approach. The transfer matrix approach enables the simple cascading of matrices through simple matrix multiplication.
The transfer matrix approach is analytically exact, and “arbitrary” heterostructures can apparently be handled through the discretization of potential changes. The approach appears to be quite appealing. However, the approach is inherently unstable for realistically extended devices which exhibit electrostatic band bending or include a large number of basis sets.
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