Symmetry of Single-walled Carbon NanotubesPart 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: TGroup "multiplication": t1 + t2 (sum of the translation vectors)

zoneBrillouin :ism)(homomorphtion Representa )( 2121 kttkktkt iii eee

rule" Umklapp"

moment" crystal ofon Conservati"

unless 0||

potential periodic-lattice a )()(Let

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tion)representaby dsubstitute ison (translati )()()( law sBloch'

Kkkkk

trr

rtrr kkt

kkt

V

VV

eP i

Finding the irreps of space groups1. 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 nanotubesChiral 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”)

Cp

memC

em

pme

C

mp

i

p

im

mp

i

p

2

2

m i.e., , by indexed are functions Basis

1,,1,0

: of Irreps

Linear quantum numbers

Brillouin zone of the (10,5) tube.q=70 a = (21)1/2a0 4.58 a0

text)(see rules Umklappspecialby

described as conserved"strictly not is "

)symmorphic-non is (because

:numbers quantumlinear with Difficulty

,

cellunit in the atoms ofnumber theis

in the rotations

cellunit theoflength theis ,

group line in the ons translati)(

22

m

C

m

q

operation crewsCm

ak

aTk

q

qq

q

aa

LL

Helical quantum numbers

Brillouin zone of the (10,5) tube.q=70 a = (21)1/2a0 4.58 a0

n = 5 q/n = 14

22

21

,~

~,

~

by generated )(~

),(GCD

subgroup rotational maximal

op.) screw ons(translati group" helical" )(

)( subgroupinvariant an form

rotations axis-z operations screw nsTranslatio

nn

n

anq

anq

qnr

qr

q

n

rq

nr

q

m

Cm

k

aCaTk

nnn

C

aT

CaT

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:

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)

Optical phonons at the point

Raman-active displacement patterns in an armchair nanotubeCalcutated with the Wigner projector technique