finalsoln - 1. Graphene consists of two interpenetrating...

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1. Graphene consists of two interpenetrating 2-dimensional lattices A and B in the x-y plane: a. Identify the nearest-neighbor vectors n i and construct the k -dependent LCAO tight-binding Hamiltonian for single-layer graphene in a p z orbital basis. Your 2x2 matrix should be in terms of overlap energy V pp π and the in-plane nearest- neighbor vectors . b. Show that near the K point on the edge of the Brillouin zone, where k = k K =4 π /3a ŷ , the two energy eigenvalues of the Hamiltonian are degenerate. [hint: cos( π /3)=1/2] c. Show that for small deviations from the K point in the Brillouin Zone, the energy bands are linearly dependent on Δ k = k - k K . [hint:cos(a+b)=cos(a)cos(b)- sin(a)sin(b)] d. If V pp = -3 eV and the lattice constant a=2.5 Å, calculate the group velocity for electrons in states close to the K point. What is the effective mass, and is this concept meaningful here?
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Bilayer graphene consists of two 2-dimensional graphene planes stacked together in the so-called “Bernal” ordering, where an atom in the A lattice of one plane aligns with a B atom in the other: e. Construct the 4x4 p z -basis LCAO Hamiltonian for this material in terms of overlap energies V pp π , V pp σ and the in-plane nearest-neighbor vectors. (nearest-neighbor
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This note was uploaded on 12/29/2011 for the course PHYSICS 731 taught by Professor Appelbaum during the Fall '11 term at Maryland.

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finalsoln - 1. Graphene consists of two interpenetrating...

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