Kronig - Penney model: a new solution
1997; IOP Publishing; Volume: 18; Issue: 5 Linguagem: Inglês
10.1088/0143-0807/18/5/015
ISSN1361-6404
Autores Tópico(s)Quantum and electron transport phenomena
ResumoThe one-dimensional Kronig - Penney (KP) potential consists of a periodic array of square-well potentials. The Schrödinger equation for an electron in this potential has a solution in the form of the Kronig - Penney equation (KPE), which illustrates the formation of electronic energy bands. The KPE is routinely found from the determinant of a matrix resulting from four boundary conditions on the wavefunction and its derivative. Here, a less tedious approach is pursued, one that is more readily adapted to multilayer structures. The notational simplicity of the present formulation pays off when an alternative form of the KPE is derived. It is shown that the new form has several conceptual, pedagogic and numerical advantages over the standard KPE. Among them, in the limit of infinite well-to-well separation, the present solution readily reduces to the solution for a single square well; it readily provides the analytic relations for the top and bottom of an energy band; for energies below the top of the barrier, it is only on the order of unity, whereas the KPE can vary over tens of orders of magnitude.
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