Symmetry of Single-walled Carbon Nanotubes Part II Outline Part II (December 6) Irreducible representations Symmetry-based quantum numbers Phonon symmetries M. Damjanovi, I. Miloevi, T. Vukovi, and J. Maultzsch, Quantum numbers and band topology of nanotubes, J. Phys. A: Math. Gen. 36, 5707-17 (2003) 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 Illustration: Electronic states in crystals Lattice translation group: T
Group "multiplication": t1 + t2 (sum of the translation vectors) Representation (homomorphism) : eikt1 eikt 2 eik ( t1 t 2 ) k Brillouin zone Bloch' s law Pt k (r ) k (r t ) eikt k (r ) (translati on is substituted by representa tion) Example of a selection rule : Let V (r ) V (r t ) a lattice - periodic potential k | V | k 0 unless k k K " Conservati on of crystal moment" " Umklapp rule" Finding the irreps of space 1. Choose a set of basis functions that span the Hilbert space of the problem groups 2. 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. 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 Schnfliess notation) Achiral nanotubes: 2n /mmm (D2nh in Schnfliess notation) Symmetry-based quantum (k ,k ) in graphene (k,m) in nanotube numbers x y k : translation along tube axis ("crystal momentum") m : rotation along cube axis ("crystal angular momentum) Irreps of C p :
Cp e i 2 m p m 0, 1, , p 1 Basis functions are indexed by m , i.e., m e im C p m e i 2
m p m Linear quantum numbers k T (a ) translations in the line group k , a a a is the length of the unit cell m Cq rotations in the screw operation
q is the number of atoms in the unit cell m q2 , q2 Difficulty with linear quantum numbers : Cq L (because L is non - symmorphic) " m is not strictly conserved" as described by special Umklapp rules (see text) Brillouin zone of the (10,5) tube. q=70 a = (21)1/2a0 4.58 a0 Helical quantum numbers Translations screw operations z - axis rotations
form an invariant subgroup Tqr ( a ) Cn Tqr (a ) " helical group" (translations screw op.) Cn maximal rotational subgroup n GCD(n1 , n2 ) ~ k Tqr (a ) generated by Cqr qn a ~ k qn a , qn a ~ C m
n ~ n , n m 2 2 Brillouin zone of the (10,5) tube. q=70 a = (21)1/2a0 4.58 a0 n = 5 q/n = 14 Irreps of nanotube line groups Translations and z-axis rotations leave |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 Damjanovi notations: Optical phonons at the point (|00): G = point group point 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) Raman-active displacement patterns in an armchair nanotube Calcutated with the Wigner projector technique
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