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Symmetry of Single-walled Carbon Nanotubes Part II

Symmetry of Single-walled Carbon Nanotubes Part II. Outline. Part II (December 6) Irreducible representations Symmetry-based quantum numbers Phonon symmetries

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Symmetry of Single-walled Carbon Nanotubes Part II

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  1. Symmetry of Single-walled Carbon NanotubesPart II

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

  3. 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 = Sym[H]  for all eigenstates | of H | = |i Vi (eigenstates can be labeled with the irrep they belong to, "quantum number")  i | j   ijselection rules

  4. Illustration: Electronic states in crystals Lattice translation group: TGroup "multiplication": t1 + t2 (sum of the translation vectors)

  5. 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.

  6. 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)

  7. 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

  8. Linear quantum numbers Brillouin zone of the (10,5) tube. q=70 a = (21)1/2a0 4.58 a0

  9. Helical quantum numbers Brillouin zone of the (10,5) tube. q=70 a = (21)1/2a0 4.58 a0 n = 5 q/n = 14

  10. 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 , |–k–m  1d (special points) and 2d irreps Achiral tubes: |km |k–m |–km |–k–m 1, 2, and 4d irreps Damjanović notations:

  11. 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)

  12. Raman-active displacement patterns in an armchair nanotube Calcutated with the Wigner projector technique

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