Kronig-Penney Model with Dirac Comb
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The simplest modification of the Kronig–Penney model for electrons in a one-dimensional periodic lattice can be based on a Dirac-comb potential approximating the positive cores:
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Contributed by: S. M. Blinder (August 2022)
Open content licensed under CC BY-NC-SA
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In the interval , the electron is a free particle and its wavefunction can be written
.
For the cell immediately to the left of the origin, with , Bloch's theorem implies that
.
The wavefunction must be continuous at , which leads to
(condition 1).
The derivatives at are found to be
and .
Taking account of the discontinuity in the first derivatives, the second derivative can be written
,
which cancels the delta function in the potential when
(condition 2).
Eliminating and between the two conditions gives
.
This can be solved for , from which it then follows that
.
References
[1] MIT OpenCourseWare. "Band Theory of Solids" (Feb 3, 2022) ocw.mit.edu/courses/chemistry/5-62-physical-chemistry-ii-spring-2008/lecture-notes/26_562ln08.pdf.
[2] S. Rajendran. "Understanding Band Structures in Solids via Solving Schrödinger Equation for Dirac Comb." (Feb 3, 2022) saravananrajendran.weebly.com/uploads/1/0/3/9/103971060/dirac_comb.pdf.
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