By Stephanie Reich, Christian Thomsen, Janina Maultzsch
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Additional info for Carbon Nanotubes: Basic Concepts and Physical Properties
SAAis found in exactly the same way as H M . It is also constant, and we set it equal to one. In other words, we assume the wave runctions V A to be normali~ed( ( ~ /(i r- RA)1 qA( r - R A ) )= I. The matrix element between A and B atoms is given by the formal expression The second sum runs over all three nearest neighbors of a given alom A. The vcctors R I i = R B -~RA ~ (i = I , 2 , 3 ) pointing from A to one of its neighbors Bli can be found from Fig. 2 We insert Eq. 13) into Eq. 12) and sum over the B neighbors and A atoms.
The degenerate eigenmode is obtained from a given eigenvector by a 90" rotation of the displacement pattern around the tube. If C4 is not a symmetry operation of the nanotube the displacement vectors must be extended to the new atomic positions. The resulting eigenvector has its nodes where the atoms are fully displaced in Fig. 14 and vice versa. The pair of projection operators P2(2)and P2(1)project nntcl another degenerate E2g mode, namely, an in-phase displacenlent of the two graphene sublattices.
However, the energies Eiin Eq. 8) depend only on products of the matrix and ovcrlap elcments with a complcx conjugate of thcsc clcments. For example, from Eq. 8) we obtain for E2 Similar cosinc terms appear in all other E, and, also, for the expressions for more distant neighbors. al + 2 c o s k . a 2 + 2 c o s k . ( a l -a2) and f(k) = 3 + u ( k ) . 17) The wave vectors k of graphene are commonly given in units of the reciprocal lattice vectors kl and k2 or in a Cartesian coordinate system in reciprocal space.
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