Symmetry of Singlewalled Carbon Nanotubes Part II

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Symmetry of Single-walled Carbon NanotubesPart II
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Outline

• Part II (December 6)
• Irreducible representations
• Symmetry-based quantum numbers
• Phonon symmetries
• M. Damjanovic, I. Miloevic, T. Vukovic, and J.
  Maultzsch, Quantum numbers and band topology of
  nanotubes, J. Phys. A Math. Gen. 36, 5707-17
  (2003)

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Application of group theory to physics
Representation ? G ?? P homomorphism to a
group of linear operators on a vector space V
(in physics V is usually the Hilbert space of
quantum mechanical states). If there exists a V1
? V invariant real subspace ? ? is reducible
otherwise it is irreducible. V can be decomposed
into the direct sum of invariant subspaces
belonging to the irreps of G V V1 ? V2 ? ?
Vm If G SymH ? for all eigenstates ?? of H
?? i? ? Vi (eigenstates can be labeled
with the irrep they belong to, "quantum
number") ? i j ? ? ?ij ? selection
rules
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Illustration Electronic states in crystals
Lattice translation group TGroup
"multiplication" t1 t2 (sum of the
translation vectors)
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Finding the irreps of space groups

• Choose a set of basis functions that span the
  Hilbert space of the problem
• Find all invariant subspaces under the symmetry
  group(Subset of basis functions that transfor
  between each other)
• Basis functions for space groups Bloch functions
  Bloch functions form invariant subspaces under
  T? only point symmetries need to be considered

• "Seitz star" Symmetry equivalent k vectors in
  the Brillouin zone of a square lattice
• 8-dimesional irrep
• In special points "small group representations
  give crossing rules and band sticking rules.

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Line groups and point groups of carbon nanotubes
Chiral nanotubs Lqp22 (q is the number of
carbon atoms in the unit cell) Achiral
nanotubes L2nn /mcm n GCD(n1, n2) ?
q/2 Point groups Chiral nanotubs q22 (Dq in
Schönfliess notation) Achiral nanotubes 2n /mmm
(D2nh in Schönfliess notation)

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Symmetry-based quantum numbers
(kx,ky) in graphene ? (k,m) in nanotube k
translation along tube axis ("crystal momentum")
m rotation along cube axis ("crystal angular
momentum)

Cp
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Linear quantum numbers

Brillouin zone of the (10,5) tube. q70 a
(21)1/2a0 ? 4.58 a0?
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Helical quantum numbers

Brillouin zone of the (10,5) tube. q70 a
(21)1/2a0 ? 4.58 a0? n 5 q/n 14
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Irreps of nanotube line groups
Translations and z-axis rotationsleave km?
states invariant. The remaining symmetry
operations U and ? Seitz stars of chiral
nanotubes km? , km? ? 1d (special points)
and 2d irreps Achiral tubes km? km? km?
km? ? 1, 2, and 4d irreps Damjanovic
notations

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Optical phonons at the ? point

• ? point (00?) G point group
• The optical selection rules are calculated as
  usual in molecular physics
• Infraded active
• A2u 2E1u (zig-zag)
• 3E1u (armchair)
• A2 5E1 (chiral)
• Raman active
• 2A1g 3E1g 3E2g (zig-zag)
• 2A1g 2E1g 4E2g (armchair)
• 3A1 5E1 6E2 (zig-zag)

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Raman-active displacement patterns in an armchair
nanotube

• Calcutated with the Wigner projector technique