We extend previous work, applying elementary matrix mechanics to one-dimensional periodic arrays (to generate energy bands), to two-dimensional arrays. We generate band structures for the square-lattice “2D Kronig-Penney model” (square wells), the “muffin-tin” potential (circular wells), and Gaussian wells. We then apply the method to periodic arrays of more than one atomic site in a unit cell, specifically to the case of materials with hexagonal lattices like graphene. These straightforward extensions of undergraduate-level calculations allow students to readily determine band structures of current research interest.
REFERENCES
For example, in Ref. 7 on p. 140 in the paragraph immediately following Eq. (8.49) there is a brief description of what is required to compute band structure, with no guide as to how to proceed technically. In the first problem at the end of that chapter, the student is guided through an analytical solution of the Kronig-Penney model, with no connections made to the paragraph following Eq. (8.49). Reference 3 makes this connection, and the present paper illustrates how to move on beyond one dimension, where no analytical solution exists.
See Ref. 7, pp. 181–182.
