Supercell-zone folding transformation for bulk crystals and nanotubes
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The analytical relation between k-points in the primitive-cell Brillouin zone and reduced supercell Brillouin zone is reported for supercell-zone-folding transformation. Examples are given for symmetry points of square and cubic simple, face-centered and body-centered lattices. The cyclic cluster symmetry is considered as a particular case of supercell-zone-folding transformation. The results of the first principles calculations of LiCl crystal in the supercell model as well as the symmetry of one-electron states are discussed using the supercell-zone-folding concept. The first principles calculations of copper impurity in LiCl crystal are made using different supercells. The site symmetry method is applied to find the space group representations induced by Cl p-states and copper d-states. The zone-folding transformation for the two-dimensional layer unit cell is considered in relation to the single wall nanotubes modeling. The results of zone-folding method application to electron states of WS2-based nanotubes and to phonon calculations of carbon and ZrS2-based nanotubes are presented and discussed.
KeywordsSymmetry analysis Unit cell transformation Brillouin zone folding Special points Electronic bands Phonon frequencies Thermodynamic functions Cyclic cluster Defective crystals Nanotubes
Two of us (R.A.E and A.V.B) acknowledge the financial support provided by the Russian Foundation for Basic Research (RFBR) (Grant No. 17-03-00130-a) for calculations on thermodynamic properties of nanotubes. One of us (I.I.T.) acknowledges the financial support of the Ministry of Education and Science of the Russian Federation Project No. 3.1463.2017, for GENKPT computer code development. The authors also acknowledge the assistance of the Computer Center of Saint-Petersburg State University in the accomplishment of high-performance computations.
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