Band structure of graphene and CNT Band structure

Band structure of graphene and CNT

Band structure of graphene and CNT Graphene : Lattice : 2 - dimensional triangular lattice y축 X축 Two basis atoms

Bloch State of the π bands y축 Three nearest neighbor X축

Nearest Neighbor Approximation

Multiply on both sides

RHS = Tight Binding Approximation 에서 Nearest Neighbor Approximation이라는 것은 Within Nearest Neighbor on site only

의 nearest neighbor y축 X축

Multiply Left= Right= on both sides

Pass the Fermi point(Dirac point)

Lattice Reciprocal Lattice y축 y축 X축 X축

Band Structure of Graphene Dirac point

Band Structure of Graphene

CNT = wrapped graphene ribbon

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’

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. , PRB 74, 165310(2006), Kane and Mele, PRL, 95, 226801(2005)

Full 4 component or 8 component solution A bit complicated

Diagonalize in real spin space Min et al. , PRB 74, 165310(2006), Kane and Mele, PRL, 95, 226801(2005)

Diagonalize in real spin space

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