Construction of Brillouin Zones

CONSTRUCTION OF BRILLOUIN ZONES

Brillouin Zones are constructed first by drawing vectors from the origin to all the lattice points and then by drawing planes which bisects these vectors perpendicularly.

Let us consider an electron in the field of two dimensional square lattice as shown in Fig. 1.23.

Different Brillouin Zones can be constructed by the following procedure. Take O' as the origin and join it to nearest lattice points say E, F, G, H. Take the perpendicular bisectors of the lattice vectors OE, OF, OG and OH. Now, we can find that these bisectors will enclose a square ABCD known as First Brillouin Zone.

Similarly join the origin to the next nearest lattice points I, J, K, L. Take the mid points M, N, P, Q and draw perpendicular to these points. These perpendicular will enclose a square EFGH. The area between EFGH and ABCD represents the Second Brillouin Zone. By the same procedure the third, fourth etc Brillouin Zones can be constructed.

Note: Brillouin Zone for an electron in a one dimensional mono atomic lattice will be a straight line as shown in fig. 1.23(a).