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

Graphene layer latticeGraphene layer lattice

Graphene 2D band Graphene 2D band structurestructure

Tigh binding first neighbor

Ab-initio

Rolling vector Rolling vector C Chh = na = na11 +ma +ma22

Ch = 3a1 +2a2

Different rollingsDifferent rollings

armchair zigzag general

Nanotube Nanotube bandstructurebandstructure

IF (n-m)/3 THEN metallic ELSE semiconductor

Metallic tube 1D band Metallic tube 1D band structurestructure

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

Effect of bundlesEffect of bundles

Band shifts Van-Hove singularities:- peak splitting and- decreased intensity