2. 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

3. Outline j Introduction -- zone folding jEffect of curvature -- jellium model results jEffect of curvature -- small energy: secondary gap jEffect of curvature -- hybridization of Pi and Sigma states: shift of the bands jBundles -- low energy: secondary gap jBundles -- visible energy range: shift of the bands

4. Graphene layer lattice

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

6. Rolling vector C h = na 1 +ma 2 C h = 3a 1 +2a 2

7. Different rollings armchair zigzag general

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

9. Metallic tube 1D band structure

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

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

12. 2. effect of rolling: Pi - Sigma mixing Graphene: pure sp 2 bonds Nanotube: mixed sp 2 and sp 3 bonds

13. Effect of curvature in simple jellium NT model Zone foldingCylindrical geometry

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

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

16. First neighbor tight binding -- inequivalent neighbors Use different  1 and  2 interaction energy for the two types of neighbors! zigzagarmchair Only armchair tubes remain true metallic because shift in k z is parallel to allowed line Zigzag: finite gap opens!

17. Curvature induced gap at E F armchair points Calculated secondary gap in quasi-metallic nanotubes STS measurement 1/d 2 gap energy

18. Sigma - Pi hybridization (5,0) NT m=0 eigenfunction at the Gamma point Most of the wave function concentrated outside the NT Strong component with equal sign on both sides

19. Curvature effect on armchair tube Zone folding is good for armchair tubes level anticrossing

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

21. 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

22. Nanotube bundles (or ropes) HRTEM image of NT ropeHexagonal packing of (10,10) tubes

23. Effect of proximity: pseudogap Hexagonal packing of (10,10) tubes Calculated DOS

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