Beyond zone folding: effect of curvature and NT bundles Márk Géza István MTA Research Institute...
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Transcript of Beyond zone folding: effect of curvature and NT bundles Márk Géza István MTA Research Institute...
Beyond zone foldingBeyond zone folding: effect : effect of curvature and NT bundlesof curvature and NT bundles
MárkMárk Géza István Géza István
MTA Research Institute forTechnical Physics and
Materials ScienceBudapest
http://www.nanotechnology.hu
OutlineOutline Introduction -- zone foldingIntroduction -- zone folding
Effect of curvature -- jellium model resultsEffect of curvature -- jellium model results
Effect of curvature -- small energy: secondary gap Effect of curvature -- small energy: secondary gap
Effect of curvature -- hybridization of Pi and Sigma Effect of curvature -- hybridization of Pi and Sigma states: shift of the bandsstates: shift of the bands
Bundles -- low energy: secondary gapBundles -- low energy: secondary gap
Bundles -- visible energy range: shift of the bandsBundles -- visible energy range: shift of the bands
NT density of states and NT density of states and tunneling spectroscopytunneling spectroscopy
STS
J. W. Wildöer et al., Nature 391 (1998) 59
1. effect of rolling: 1. effect of rolling: inequivalent neighborsinequivalent neighbors
Rolling
a
d<a
Along the circumference: bond length and angle changeAlong the NT axis: no change
2. effect of rolling:2. effect of rolling:Pi - Sigma mixingPi - Sigma mixing
Graphene: pure sp2 bondsNanotube: mixed sp2 and sp3 bonds
Effect of curvature in Effect of curvature in simple jellium NT modelsimple jellium NT model
Zone folding Cylindrical geometry
Jellium solutionsJellium solutions
Zone folding:superposition ofplane waves
Cylindrical geometry:superposition ofBessel functions
Effect of NT radiusEffect of NT radius
L.Tapasztó et. al. , AIP Conf.Proc., 685, 439, (2003)
E(r;m) functions E(m;r) functions
First neighbor tight First neighbor tight binding -- inequivalent binding -- inequivalent
neighborsneighborsUse different 1 and 2 interaction energy
for the two types of neighbors!
zigzag armchair
Only armchair tubes remaintrue metallic because shiftin kz is parallel to allowedline
Zigzag: finite gap opens!
Curvature induced gap at Curvature induced gap at EEFF
armchair points
Calculated secondary gapin quasi-metallic nanotubes
STS measurement1/d2 gap energy
Sigma - Pi hybridizationSigma - Pi hybridization
(5,0) NT m=0 eigenfunctionat the Gamma point
• Most of the wave function concentrated outside the NT• Strong component with equal sign on both sides
Curvature effect onCurvature effect onarmchair tubearmchair tube
• Zone folding is good for armchair tubes
level anticrossing
Curvature effect: zigzag Curvature effect: zigzag tubetube
• Pi* bands shifted down• Sigma* bands shifted up
Effect of curvatureEffect 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 Nanotube bundles (or ropes)ropes)
HRTEM image of NT rope Hexagonal packingof (10,10) tubes
Effect of proximity: Effect of proximity: pseudogappseudogap
Hexagonal packingof (10,10) tubes Calculated DOS