Band structure of graphene and CNT

Graphene : Lattice : 2- dimensional triangular lattice Two basis atoms X축X축 y축y축

Bloch State of the π bands X축X축 y축y축 Three nearest neighbor

Nearest Neighbor Approximation

Multiply on both sides

RHS = Tight Binding Approximation 에서 Nearest Neighbor Approximation 이라는 것은 Within Nearest Neighboron site only

X축X축 y축y축 의 nearest neighbor

Multiply on both sides Left= Right=

Pass the Fermi point(Dirac point)

X축X축 y축y축 X축X축 y축y축 LatticeReciprocal Lattice

Band Structure of Graphene Dirac point

Band Structure of Graphene

“ - 공간 에서의 주기함수 ” 를 확인하시기 바랍니다.

CNT = wrapped graphene ribbon

X축X축 경계조건 y축y축

For example,

Subband (n=0) Subband (n=1)

Low energy effective Hamiltonian near K and K’

Tight-binding π bands, again.

Near K or K’

spinor of pseudospin

Mahmut, you have the solution for the spinor Bands are doubly degenerate in real spin

With SOC

In this low-energy Cone region, how and why the SOC is represented this way? Min et al., PRB74,165310(2006), Kane and Mele, PRL, 95, (2005)

Full 4 component or 8 component solution  A bit complicated

Diagonalize in real spin space Min et al., PRB74,165310(2006), Kane and Mele, PRL, 95, (2005)

Diagonalize in real spin space

Effective Hamiltonian Including the two Fermi point K and K’ (K’=-K) Without SOC it is not very meaningful

Effective Hamiltonian Including the two Fermi point K and K’ (K’=-K) Why do we need this ? ????

Why do we need this ?

Near K

Near K’=-K