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Beyond zone folding : effect of curvature and NT bundles. Márk Géza István MTA Research Institute for Technical Physics and Materials Science Budapest. http://www.nanotechnology.hu. Outline. Introduction -- zone folding Effect of curvature -- jellium model results
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Beyond zone folding: effect of curvature and NT bundles Márk Géza István MTA Research Institute for Technical Physics and Materials Science Budapest http://www.nanotechnology.hu
Outline • Introduction -- zone folding • Effect of curvature -- jellium model results • Effect of curvature -- small energy: secondary gap • Effect of curvature -- hybridization of Pi and Sigma states: shift of the bands • Bundles -- low energy: secondary gap • Bundles -- visible energy range: shift of the bands
Ab-initio Graphene 2D band structure Tigh binding first neighbor
Rolling vector Ch = na1 +ma2 Ch = 3a1 +2a2
armchair zigzag general Different rollings
Nanotube bandstructure IF (n-m)/3 THEN metallic ELSE semiconductor
NT density of states and tunneling spectroscopy STS J. W. Wildöer et al., Nature 391 (1998) 59
1. effect of rolling: inequivalent neighbors a d<a Rolling Along the circumference: bond length and angle changeAlong the NT axis: no change
2. effect of rolling:Pi - Sigma mixing Graphene: pure sp2 bonds Nanotube: mixed sp2 and sp3 bonds
Effect of curvature in simple jellium NT model Zone folding Cylindrical geometry
Jellium solutions Zone folding: superposition of plane waves Cylindrical geometry: superposition of Bessel functions
Effect of NT radius E(r;m) functions E(m;r) functions L.Tapasztó et. al. , AIP Conf.Proc., 685, 439, (2003)
First neighbor tight binding -- inequivalent neighbors Use different g1 and g2 interaction energyfor the two types of neighbors! Only armchair tubes remaintrue metallic because shiftin kz is parallel to allowed line Zigzag: finite gap opens! zigzag armchair
Curvature induced gap at EF Calculated secondary gapin quasi-metallic nanotubes armchair points STS measurement1/d2 gap energy
Sigma - Pi hybridization • Most of the wave function concentrated outside the NT • Strong component with equal sign on both sides (5,0) NT m=0 eigenfunctionat the Gamma point
Curvature effect onarmchair tube • Zone folding is good for armchair tubes level anticrossing
Curvature effect: zigzag tube • Pi* bands shifted down • Sigma* bands shifted up
Effect of curvature • Valence bands not affected by curvature • Two-fold degeneracy of Sigma* is lifted in NT • Armchair Sigma* does not change • Pi* band moves down • Sigma* band moves up
Nanotube bundles (or ropes) HRTEM image of NT rope Hexagonal packingof (10,10) tubes
Effect of proximity: pseudogap Hexagonal packingof (10,10) tubes Calculated DOS
Effect of bundles Band shifts Van-Hove singularities:- peak splitting and- decreased intensity