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Pick four out of six problems below. Students who solve all six problems will be given extra credit.

Problem 1

The energy momentum-dispersion of graphene valence and conduction bands is given by the formula derived in class:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(k_x,k_y) = \pm \sqrt{3 + 4 \cos \left( \frac{\sqrt{3}}{2} k_y a \right) \cos\left(\frac{3}{2}k_xa\right) + 2 \cos(\sqrt{3}k_ya)} }

which assumes tight-binding Hamiltonian projected to a basis set of single Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \pi } orbital per carbon atom and hopping allowed only between the nearest neighbor carbon atoms. The honeycomb lattice spacing is Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a } .

(a) Using MATLAB or Mathematica plot this function as a surface in 3D k-space within the range of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (k_x,k_y) } vector values defined by the first Brillouin zone.

(b) Near one of the two inequivalent K points [the so-called Dirac points at which Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(k_x,k_y) \equiv 0 } ] in the corners of the first Brillouin zone, where the conduction (top surface) and valence (bottom surface) bands touch, the dispersion has a conical shape characteristics of massless relativistic particles (such as photons or neutrinos). Up to what energy away from the Dirac point Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E=0 } (which is the Fermi energy of ideal undoped graphene) can the real dispersion shown in your figure be approximated by linear dispersion of the "Dirac cones"? You can subtract Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(k_x,k_y) } surface around one of the K points from the conical surface Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E(q_x, q_y) = \pm v_F \hbar \sqrt{q_x^2+q_y^2} } (corrections are Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle O[(q/K)]^2} ; here Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{q} = \mathbf{K} + \mathbf{k} } and the Fermi velocity is Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle v_F = \frac{\sqrt{3}}{2} t a/\hbar \approx 10^6 m/s } ) and find at which energy Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \pm E_0 } their difference starts to be more than 5%.

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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{H}_0 } describing graphene supercells coupled to adjacent supercells via the hopping submatrices Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t } .

(a) Using the honeycomb (or equivalent brick) lattice of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 50 \times 50 } carbon atoms, construct the full Hamiltonian matrix Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle [-3.5t, 3.5t]} .

(b) Introduce periodic boundary into Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{H} } (this requires to input two additional submatrices Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{H}_1 } into Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{H} } , as well as to introduce extra hoppings Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t } into Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \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, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t } and the asymmetry parameter Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta = \frac{1}{2} \left(V_A - V_B \right) } .

b) Linearize the solution found in part a) near the points Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle K } and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle K^\prime } treating the asymmetry as a small perturbation, and find how the massless Dirac picture (valid for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V_A = V_B = 0 } ) of low energy states of graphene is altered. That is, show that Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta } generates a finite mass of Dirac quasiparticles whose Hamiltonian is given by:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{H} = v_F \hat{\mathbf{\sigma}} \cdot \hat{\mathbf{p}} + \lambda \hat{\sigma}_z } ,

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda = V_A = - V_B } .