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Brillouin zones are an important characteristic of crystal structures. The construction and illustration of Brillouin zones for a three dimensional lattice are somewhat difficult to follow. The construction of Brillouin zones for a two dimensional lattice is much easier to follow.
This is a sketch of the construction of the first four Brillouin zones for a square lattice. First, some definitions are needed.
A Bragg plane for two points in a lattice is the plane which is perpendicular to the line between the two points and passes through the bisector of that line. The first Brillouin zone for a point in a lattice is the set of points that are closer to the point than the Bragg plane of any point. In other words one can reach any of the points in the first Brillouin zone of a lattice point without crossing the Bragg plane of any other point in the lattice.
The second Brillouin zone is defined as the points which may be reached from the first Brillouin zone by crossing only one Bragg "plane." This can be generalized to define the n-th Brillouin zone as the set of points, not in the previous zones, that can be reached from one (n-1)-th zone by crossing one and only one Bragg plane.
In constructing the Brillouin zones for a point it is expedient to first determine the nearest neighbors, the next nearest neighbors and so on. This is conveniently illustrated with a square lattice. Shown below are the nearest through fourth-nearest neighbors and their Bragg lines.
The zones can easily be determined from their definitions. The first zone, one within all of the Bragg lines is shown red below. The second zone is all the points that can be reached by crossing one and only one Bragg line from the first zone. The second zone is shown in green in the illustration below. The third zone, shown in blue, consists of all the points that can be reached by crossing only one Bragg line from the second zone. The fourth zone is shown in black.
Consider a two-dimensional lattice which has two electrons per unit cell.
The area of the Fermi circle for the electrons then has an area
equal to the area of the first Brillouin zone, 1(lattice unit)2.
The radius of the Fermi circle is thus
1/(π)1/2. The Fermi circle superimposed on the Brillouin zones is as follows:
The components of the Fermi circle outside of the first Brillouin zone may be translated back into the first zone by a lattice vector. The result is as shown below.
When there are four electrons per unit cell the Fermi has twice the area; i.e., 2, and the radius of the Fermi circle is then [2/π]1/2. This Fermi circle superimposed upon the Fermi zones is as shown below:
When all components of the Fermi circle are translated back to the first Brillouin zone the result is as below:
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