finalsoln

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

This preview shows pages 1–4. Sign up to view the full content.

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?

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 12/29/2011 for the course PHYSICS 731 taught by Professor Appelbaum during the Fall '11 term at Maryland.

### Page1 / 9

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online