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Band structure of graphene and CNT
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Graphene : Lattice : 2- dimensional triangular lattice Two basis atoms X축X축 y축y축
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Bloch State of the π bands X축X축 y축y축 Three nearest neighbor
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Nearest Neighbor Approximation
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Multiply on both sides
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RHS = Tight Binding Approximation 에서 Nearest Neighbor Approximation 이라는 것은 Within Nearest Neighboron site only
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X축X축 y축y축 의 nearest neighbor
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Multiply on both sides Left= Right=
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Pass the Fermi point(Dirac point)
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X축X축 y축y축 X축X축 y축y축 LatticeReciprocal Lattice
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Band Structure of Graphene Dirac point
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Band Structure of Graphene
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“ - 공간 에서의 주기함수 ” 를 확인하시기 바랍니다.
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CNT = wrapped graphene ribbon
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X축X축 경계조건 y축y축
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For example,
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Subband (n=0) Subband (n=1)
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Low energy effective Hamiltonian near K and K’
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Tight-binding π bands, again.
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Near K or K’
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spinor of pseudospin
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Mahmut, you have the solution for the spinor Bands are doubly degenerate in real spin
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With SOC
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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)
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Full 4 component or 8 component solution A bit complicated
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Diagonalize in real spin space Min et al., PRB74,165310(2006), Kane and Mele, PRL, 95, 226801(2005)
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Diagonalize in real spin space
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Effective Hamiltonian Including the two Fermi point K and K’ (K’=-K) Without SOC it is not very meaningful
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Effective Hamiltonian Including the two Fermi point K and K’ (K’=-K) Why do we need this ? ????
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Why do we need this ?
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Near K
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Near K’=-K
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