Beyond zone folding: effect of curvature and NT bundles

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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
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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

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Graphene layer lattice
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Graphene 2D band structure
Tigh binding first neighbor
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Rolling vector Ch na1 ma2
Ch 3a1 2a2
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Different rollings
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Nanotube bandstructure
IF (n-m)/3 THEN metallic ELSE
semiconductor
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Metallic tube 1D band structure
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NT density of states and tunneling spectroscopy
STS J. W. Wildöer et al., Nature 391 (1998) 59
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1. effect of rolling inequivalent neighbors
a
Rolling
dlta
Along the circumference bond length and angle
changeAlong the NT axis no change
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2. effect of rollingPi - Sigma mixing
Graphene pure sp2 bonds Nanotube mixed sp2 and
sp3 bonds
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Effect of curvature in simple jellium NT model
Zone folding
Cylindrical geometry
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Jellium solutions
Zone folding superposition of plane waves
Cylindrical geometry superposition of Bessel
functions
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Effect of NT radius
E(rm) functions
E(mr) functions
L.Tapasztó et. al. , AIP Conf.Proc., 685, 439,
(2003)
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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
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Curvature induced gap at EF
Calculated secondary gapin quasi-metallic
nanotubes
armchair points
STS measurement1/d2 gap energy
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Sigma - Pi hybridization

• Most of the wave function concentrated outside
  the NT
• Strong component with equal sign on both sides

(5,0) NT m0 eigenfunctionat the Gamma point
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Curvature effect onarmchair tube

• Zone folding is good for armchair tubes

level anticrossing
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Curvature effect zigzag tube

• Pi bands shifted down
• Sigma bands shifted up

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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

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Nanotube bundles (or ropes)
HRTEM image of NT rope
Hexagonal packingof (10,10) tubes
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Effect of proximity pseudogap
Hexagonal packingof (10,10) tubes
Calculated DOS
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Effect of bundles
Band shifts
Van-Hove singularities- peak splitting and-
decreased intensity