Kronig-Penney Model with Dirac Comb

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:
.
Here, is the lattice spacing for an infinite row of delta functions. We consider the case of a repulsive potential, with . The Schrödinger equation (in atomic units ),
, with ,
can be solved exactly by applying Bloch's theorem, such that
,
where is the crystal momentum. As derived in the Details section, the allowed energies are then determined by the transcendental equation
.
Since , only values of fulfilling this condition are allowed. Selecting "graphical solution" shows these values of occupying a sequence of bands, shown in blue. This simple model thus simulates the electronic band structure of solids, which is essential for the understanding of insulators, conductors and semiconductors.
Select "energy bands" to see an idealized representation of the band structure, showing the first few energy bands separated by gaps. Completely filled bands generally make the material an insulator, since it takes considerable energy to excite an electron across a gap. However, if at least one band is partially filled, it takes less energy to excite an electron, and this material is typically a conductor. An insulator selectively doped with a few atoms can possibly create extra electrons or holes in a filled band, which results in a semiconductor, through which weak currents can flow.

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DETAILS

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
[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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