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The Kronig-Penney model extended to arbitrary potentials via numerical matrix mechanics

The Kronig-Penney model describes what happens to electron states when a confining potential is repeated indefinitely. This model uses a square well potential; the energies and eigenstates can be obtained analytically for a the single well, and then Bloch's Theorem allows one to extend these solutions to the periodically repeating square well potential. In this work we describe how to obtain simple numerical solutions for the eigenvalues and eigenstates for any confining potential within a unit cell, and then extend this procedure, with virtually no extra effort, to the case of periodically repeating potentials. In this way one can study the band structure effects which arise from differently-shaped potentials. One of these effects is the electron-hole mass asymmetry. More realistic unit cell potentials generally give rise to higher electron-hole mass asymmetries.