1. Band structure of graphene and CNT

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

3. Bloch State of the π bands X축X축 y축y축 Three nearest neighbor

4. Nearest Neighbor Approximation

5. Multiply on both sides

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

7. X축X축 y축y축 의 nearest neighbor

8. Multiply on both sides Left= Right=

10. Pass the Fermi point(Dirac point)

12. X축X축 y축y축 X축X축 y축y축 LatticeReciprocal Lattice

13. Band Structure of Graphene Dirac point

14. Band Structure of Graphene

15. “ - 공간 에서의 주기함수 ” 를 확인하시기 바랍니다.

17. CNT = wrapped graphene ribbon

18. X축X축 경계조건 y축y축

19. For example,

20. Subband (n=0) Subband (n=1)

23. Low energy effective Hamiltonian near K and K’

24. Tight-binding π bands, again.

25. Near K or K’

26. spinor of pseudospin

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

28. With SOC

29. 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, 226801(2005)

30. Full 4 component or 8 component solution  A bit complicated

31. Diagonalize in real spin space Min et al., PRB74,165310(2006), Kane and Mele, PRL, 95, 226801(2005)

32. Diagonalize in real spin space

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

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

36. Why do we need this ?

38. Near K

39. Near K’=-K