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Problem 2

The honeycomb lattice of graphene is topologically equivalent to a brick lattice shown in the Figure below:

File:Brick lattice.png

Honeycomb lattice of graphene (top left), its equivalent brick lattice (top right) and submatrices of the matrix representation of the tight-binding Hamiltonian defined on either of these two lattices.

Thus, the matrix representation of the Hamiltonian defined on this lattice can be obtained in the same fashion as for the square lattice (used in Problem 4. of Homework Set 2), i.e., by loading its submatrices $\mathbf {H} _{0}$ describing graphene supercells coupled to adjacent supercells via the hopping submatrices $\mathbf {H} _{1}$. The carbon atoms within the supercells as well as the structure of these submatrices are shown in the Figure. Here it is assumed that electron can hop between only the nearest neighbor carbon atoms with probability $t$.

(a) Using the honeycomb or equivalent brick lattice of $50\times 50$ sites, construct the full Hamiltonian matrix $\mathbf {H}$ of finite size graphene sheet and compute its density of states (DOS) via the Green function method used in Homework Set 2 in the energy range $[-3.5t,3.5t]$.

(b) Introduce periodic boundary into $\mathbf {H}$ (this requires to input two additional submatrices $\mathbf {H} _{1}$ into $\mathbf {H}$, as well as to introduce extra hoppings $t$ into $\mathbf {H} _{0}$) and compute its DOS via the same method as in (a).

(c) Does any of your numerically obtained DOS for finite-size graphene sheet resembles analytical DOS shown in Fig. 5 (page 114) of A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Rev. Mod. Phys. 81, 109 (2009). [PDF]?

Problem 3

Massive Dirac particles in graphene: Consider a tight-binding model on a honeycomb lattice with on-site potential1 different on the sublattices A and B, $V_{A}=-V_{B}=\lambda$.

a) Develop a general solution of this problem in terms of plane wave states, expressing the spectrum of excitations in terms of the nearest neighbor hopping amplitude $t$ and the asymmetry parameter $\Delta ={\frac {1}{2}}\left(V_{A}-V_{B}\right)$.

b) Linearize the solution found in part a) near the points $K$ and $K^{\prime }$ treating the asymmetry as a small perturbation, and find how the massless Dirac picture (valid for $V_{A}=V_{B}=0$) of low energy states of graphene is altered. That is, show that $\Delta$ generates a finite mass of Dirac quasiparticles whose Hamiltonian is given by:

${\hat {H}}=v_{F}{\hat {\mathbf {\sigma } }}\cdot {\hat {\mathbf {p} }}+\lambda {\hat {\sigma }}_{z}$,

where $\lambda =V_{A}=-V_{B}$.

Problem 4

Bilayer graphene: Suppose instead of a single layer of graphene one has two layers, stacked atop one another according to the Bernal stacking. This means that the B sublat- tice sites of the upper layer (layer 1) are directly above the A sublattice sites of the lower layer (layer 2). The A sublattice sites of the upper layer and the B sublattice sites of the lower layer have no “partner” atoms below/above them.

• (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.

• (b) Two of the bands found above touch at the Fermi energy ( = 0) at the zone boundary

     points ±K. Find the effective mass around these points.

• (c) Now calculate the Berry phase ( dk · A(k)) acquired by an electron encircling one of

     the zone boundary points in each of the two touching bands. Use your result to argue
     that the bands cannot split if a small perturbation is applied provided inversion and
     time-reversal symmetry are preserved. Where is the inversion center for the bilayer?

• (d) Check this conclusion in a simple case: Because the B1 and A2 sites have more atoms

     close by than the B2 and A1 sites, there will generally be some difference of the site
     energies for these orbitals. Add an energy +u to the electrons on the B1,A2 sites and
     −u to the electrons on the B2,A1 sites. Show that for small enough u, the two bands
     still touch.