Transcript Symmetry of Single-walled Carbon Nanotubes

Symmetry of Single-walled Carbon
Nanotubes
Part II
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)
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)
Representa tion (homomorph ism) : eikt1 eikt2  eik ( t1  t 2 ) k  Brillouin zone
Bloch' s law Pt k (r )   k (r  t )  eikt k (r )
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"
(translati on is substitute d by representa tion)
Finding the irreps of space groups
1. Choose a set of basis functions that span the Hilbert space of the problem
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 Schönfliess notation)
Achiral nanotubes:
2n /mmm (D2nh in Schönfliess notation)
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”)
Irreps of C p :
i
e
Cp
2
m
p
m  0, 1,, p  1
Basis functions are indexed by m , i.e., m  e  im
Cp m  e
i
2
m
p
m
Linear quantum numbers
k  T (a ) translati ons 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  2, 2
q q

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
Translatio ns  screw operations  z - axis rotations
form an invariant subgroup Tqr (a )  Cn
Tqr (a ) " helical group" (translati ons  screw op.)
Cn maximal rotational subgroup
n  GCD(n1 , n2 )
~
k  Tqr (a ) generated by Cqr qn a
~
q
q
k   n a , n 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 , |–k–m
 1d (special points) and 2d irreps
Achiral tubes: |km |k–m |–km |–k–m
 1, 2, and 4d irreps
Damjanović notations:
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)
Raman-active displacement patterns in an
armchair nanotube
Calcutated with the Wigner projector technique