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Nanostructure Curvature Effects and Nanotube Bundles: Insights from Zone Folding

Explore the impact of curvature on nanotubes through zone folding and the effects of bundling on band structures, discussing jellium models, energy shifts, and hybridization states.

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Nanostructure Curvature Effects and Nanotube Bundles: Insights from Zone Folding

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  1. 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

  2. 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

  3. Graphene layer lattice

  4. Ab-initio Graphene 2D band structure Tigh binding first neighbor

  5. Rolling vector Ch = na1 +ma2 Ch = 3a1 +2a2

  6. armchair zigzag general Different rollings

  7. Nanotube bandstructure IF (n-m)/3 THEN metallic ELSE semiconductor

  8. Metallic tube 1D band structure

  9. NT density of states and tunneling spectroscopy STS J. W. Wildöer et al., Nature 391 (1998) 59

  10. 1. effect of rolling: inequivalent neighbors a d<a Rolling Along the circumference: bond length and angle changeAlong the NT axis: no change

  11. 2. effect of rolling:Pi - Sigma mixing Graphene: pure sp2 bonds Nanotube: mixed sp2 and sp3 bonds

  12. Effect of curvature in simple jellium NT model Zone folding Cylindrical geometry

  13. Jellium solutions Zone folding: superposition of plane waves Cylindrical geometry: superposition of Bessel functions

  14. Effect of NT radius E(r;m) functions E(m;r) functions L.Tapasztó et. al. , AIP Conf.Proc., 685, 439, (2003)

  15. 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

  16. Curvature induced gap at EF Calculated secondary gapin quasi-metallic nanotubes armchair points STS measurement1/d2 gap energy

  17. 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

  18. Curvature effect onarmchair tube • Zone folding is good for armchair tubes level anticrossing

  19. Curvature effect: zigzag tube • Pi* bands shifted down • Sigma* bands shifted up

  20. 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

  21. Nanotube bundles (or ropes) HRTEM image of NT rope Hexagonal packingof (10,10) tubes

  22. Effect of proximity: pseudogap Hexagonal packingof (10,10) tubes Calculated DOS

  23. Effect of bundles Band shifts Van-Hove singularities:- peak splitting and- decreased intensity

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