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| ==Problem 3== | | ==Problem 3== |
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| '''Bilayer graphene:''' Suppose instead of a single layer of graphene one has two layers,
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| stacked atop one another according to the ''Bernal'' stacking. This means that the B sublat-
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| tice sites of the upper layer (layer 1) are directly above the A sublattice sites of the lower
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| layer (layer 2). The A sublattice sites of the upper layer and the B sublattice sites of the
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| lower layer have no “partner” atoms below/above them.
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| *(a) Formulate and solve the tight-binding model for this system consisting of the usual hopping t between nearest-neighbor sites in each layer, and an additional smaller hopping γ between the B1-A2 sites in different layers. Choose the zero of energy to equal that of an isolated atom. You should find 4 bands.
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| *(b) Two of the bands found above touch at the Fermi energy ( = 0) at the zone boundary
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| points ±K. Find the effective mass around these points.
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| *(c) Now calculate the Berry phase ( dk · A(k)) acquired by an electron encircling one of
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| the zone boundary points in each of the two touching bands. Use your result to argue
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| that the bands cannot split if a small perturbation is applied provided inversion and
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| time-reversal symmetry are preserved. Where is the inversion center for the bilayer?
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| *(d) Check this conclusion in a simple case: Because the B1 and A2 sites have more atoms
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| close by than the B2 and A1 sites, there will generally be some difference of the site
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| energies for these orbitals. Add an energy +u to the electrons on the B1,A2 sites and
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| −u to the electrons on the B2,A1 sites. Show that for small enough u, the two bands
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| still touch.
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| ==Problem 4== | | ==Problem 4== |
