Brillouin Zone of a Single-Walled Carbon Nanotube
This Demonstration shows the construction of the first Brillouin zone (BZ) of a single-walled carbon nanotube (SWNT) superimposed on the 2D hexagonal first BZ of graphene. The first BZ of a SWNT is given by an irreducible set of equidistant lines (also called cutting-lines or 1D BZs) whose spacing and length are related to the chiral indices of the nanotube. These indices are integers that specify uniquely the geometry of the SWNT (diameter, chiral angle, length of the unit cell along the tube axis, number of graphene unit cells inside the SWNT unit cell, etc.). All the points on the cutting lines belong to the SWNT BZ, thus it is a subset of points belonging to the graphene BZ. The spacing between cutting lines is inversely proportional to the SWNT diameter, while the length of the lines is inversely proportional to the length of the SWNT unit cell along the tube axis. The orientation of the cutting lines as a function of the chiral indices depends on the SWNT chiral angle . For zigzag SWNTs , for armchair , for any chiral SWNT . The irreducible number of cutting lines is equal to the number of graphene unit cells inside the SWNT unit cell. Usually they are indexed as or equivalently by . Additionally, is cyclically periodic in . When a cutting line crosses the graphene BZ special points K and K', so that , the SWNT is metallic; otherwise, when , it is semiconducting, according to the zone-folding picture by Saito and Dresselhaus (the neglect of curvature effects is valid for large diameter tubes).
Snapshot 1: Brillouin zone of a zigzag semiconducting SWNT:
Snapshot 2: Brillouin zone of a zigzag metallic SWNT:
Snapshot 3: Brillouin zone of an armchair SWNT (always metallic)
Snapshot 4: Brillouin zone of a chiral metallic SWNT:
R. Saito, G. Dresselhaus, and M. S. Dresselhaus, Physical Properties of Carbon Nanotubes, London: Imperial College Press, 1998.
J. Alfonsi, Small Crystal Models for the Electronic Properties of Carbon Nanotubes, PhD thesis, University of Padova, 2009, Chapters 2, 3, and references therein.