What Program BZ can do (Back)

Program BZ is designed to be used mainly for exploring the Brillouin zone of a crystalline solid.  It can be used for working with the Brillouin zone for one-dimensional polymers such as polyethylene or cellulose, and for two-dimensional systems such as graphene, but the main use is for mapping the Brillouin zone of organic, organometallic, and inorganic solids. For this reason all the following discussion will be restricted to three-dimensional solids.

Structures in k-space

The volume within the Brillouin zone is referred to as k-space. Several regions in k-space are of interest, and all of these can be examined using BZ:

• Surfaces: A surface can be defined as a two-dimensional plane in k-space.  The most important of these are the various surfaces of the Brillouin zone itself, and certain cross-sections of the Brillouin zone; these cross-sections typically include the center of the Brillouin zone, i.e., k-space coordinate  (0.0, 0.0, 0.0).  Surfaces can be of any size, although most interesting features are in the first Brillouin zone. Typically, only one energy surface would be generated in a given plot.

• Lines: The shape of a Brillouin zone can be represented by a set of lines.  Those on the surface define the limits of the first Brillouin zone.  Another set of lines go from the center of the Brillouin zone to a point on the surface, either a vertex or the center of a polygon.  These lines and others, e.g., a line from the center of a polygon to a vertex, or a line that is not related to any specific feature, can be generated.  Each point on a line contains all the energy levels for that point. A set of contiguous lines will be referred to as a "walk", as in "The walk from Γ to X to W to L to Γ"  The band-structure of a solid is most commonly represented by one or more walks in k-space.

• Points.  The properties of an individual point in k-space can be calculated.  In addition to the energy levels, the eigenvalues, at the point being examined, the symmetry properties of the associated eigenvectors can also be generated.  This does require symmetry information to be supplied via a file named <file>.ops.  The symmetry group at a given point is called the "little group."  Some little groups are isomorphous with a point-group, some other little groups have no equivalent point group.

Types of energy band structure

Electronic

Electronic band structures are calculated using the Fock matrix from a MOPAC calculation.  This is the simplest type of band-structure. The number of energy bands is equal to the number of atomic orbitals in the primitive unit cell, i.e., the smallest block that can generate the solid using only translations.

Phonon

Vibrational band structures can be calculated if a FORCE calculation that includes the MERS keyword is run. The mass-weighted Hessian matrix generated in MOPAC is then written to <file>.brz.

Symmetry operations

If symmetry operations are supplied, via a file <file>.ops, then the electronic energy or phonon matrix is symmetrized.  In principle, these matrices should already have the correct symmetry, but small deviations occur due to incomplete geometry optimization and to the use of finite numerical precision in constructing the matrices, in particular the phonon matrix.  Symmetrizing the energy matrix results in the elimination of small artifacts, such as bands that should cross having a small band-gap.

When points are calculated, the characters of the symmetry operations of the associated eigenvectors are generated.  In diamond, for example, at the Γ point, each eigenvector belongs to an irreducible representation of the little group that is isomorphous with point-group Oh. At the X point, all irreducible representations are two-fold degenerate.

In general, and with the notable exception of the Identity operation, characters of eigenvectors are complex, and the imaginary component represents the momentum of the k-point.

Graphical representation

BZ is an interactive program.  While it's running, the band-structure, either a walk or a surface, is generated  so that you can see what the results look like.  This band structure can be captured using "snipping tool" or by screen capture.  The band structure is also written to a file in a form that is suitable for use by Excel.

Run in non-interactive mode

As an alternative to typing data interactively, a file can be written that contains all the typed responses.  Thus if the first response is the number "1" followed by (return), then the file would have as the first line the number 1.  Generating an aesthetically pleasing band-structure takes some trial-and-error.  If the keystrokes that were used in making such a graphic are put into a file, , and when such a