Assignment #7

Tight-binding model of the electronic band structure

1. Using the tight-binding method and assuming spherically symmetric wave functions and only nearest-neighbor interactions:

(a) (3 pt.) Find E(k) for the simple cubic, bcc, and fcc crystal structures.

(b) (1 pt.) Find E(k) for a two-dimensional square lattice.

(c) (1 pt.) Plot E(k)=const. in the first BZ in (b). Discuss the relation to the Fermi surface.

(In a, b, and c, treat the overlap integral b as a parameter.)

2. Determine the electronic band structure of silicon in different structures using the tight-binding formalism. This is a computational assignment and requires access to the PA computer kepler. All you have to do is to execute the existing program SKBAND and interpret the results.

• As a first step, on the PA Departmental computer kepler, copy the executable file skband into your own directory by issuing the command
  cp /h0/tomanek/www/PHY971/projects/skband .
  Then, copy the input files sc.skin, fcc.skin, bcc.skin, dia.skin into your directory.
• Run the code for a particular "electronic input" file, such as sc.skin for a simple cubic lattice, by issuing the command
  skband < sc.skin > sc.skout
  The results will be in the file sc.skout. A second file skbnd will be generated with information useful for plotting.

(a) (2 pt.) Which of the fcc, bcc, simple cubic and diamond structures of silicon is metallic and which is a semiconductor?

(b) (2 pt.) Find the bandwidth and the width of the band gap(s) for each structure. Is the bandwidth correlated with the coordination number?

(c) (2 pt.) Identify valence and conduction bands in the semiconducting structure(s). Find the k-points corresponding to the top of the valence band and the bottom of the conduction band. Compare to published results.