ABSTRACT
The following problems focus on the electronic and optical properties of graphene-related systems, and they can be solved by using the tight-binding model and the gradient approximation method in Chapter 2.
1.The essential properties of monolayer graphene are determined by the nearest-neighbor intralayer hopping integral of 2p z orbitals. Calculate (1) energy bands and wave functions in the first Brillouin zone; and (2) velocity matrix elements during vertical optical excitations. Also, (3) discuss the main features of the velocity matrix elements near the Dirac point; and (4) examine whether the low-energy velocity matrix elements have a constant value.
2.Consider an AA-stacked bilayer graphene with the highest configuration symmetry. When the nearest-neighboring intralayer and the vertical interlayer hopping integrals are taken into account, show that (1) two pairs of Dirac cones can be characterized by analytic forms; (2) only the intra-Dirac-cone vertical transitions are available in absorption spectra by examining the linear superposition of the four tight-binding functions. (3) Show that the main features of Dirac cones in (1) and (2) are not affected by the gate voltage (a uniform perpendicular electric field).
3.Based on the low-energy expansions near the K point, (1) obtain the approximate Hamiltonian matrix elements for monolayer and AA bilayer stacking. Use creation and destruction operators to determine (2) the low-lying Landau levels (LLs) and well-behaved spatial distributions; (3) the specific magneto-optical selection rule; and (4) the threshold absorption frequency.
4.Monolayer graphene is present in a non‑uniform perpendicular magnetic field. This field is assumed to be spatially modulated along the armchair direction in the cosine form, leading to an enlarged unit cell with 2N B carbon atoms. Calculate (1) the vector potential; (2) the near-neighbor Peierls phases; and (3) the independent hopping integrals or Hamiltonian matrix elements. Similar calculations should be generalized to a uniform magnetic field accompanied by a modulated (4) electric or (5) magnetic field.
5.(1) Discuss the similarities and differences between AA-stacked graphenes and simple graphite for electronic and optical properties in the absence/presence of a uniform perpendicular magnetic field. (2) Apply the same investigations for AB-stacked graphenes and Bernal graphite.122
6.For (1) trilayer ABA-stacked graphenes and (2) ABC-stacked graphenes, calculate the Hamiltonian matrix elements under a uniform magnetic field using the intralayer and interlayer hopping integrals in Chapter 2.
7.Explain why it is very difficult to observe the dimensional crossover behavior associated with ABC-stacked graphenes and rhombohedral graphite.
8.Carbon nanotubes possess the periodical boundary condition and the curvature effect. Evaluate the analytic π-electronic energy dispersions for (1) (P,P) armchair nanotubes; and (2) (P,0) zigzag nanotubes by using the nearest-neighbor hopping integrals provided in Section 6.1. Show that (3) the magneto-electronic band structures exhibit the Aharonov–Bohm effect in the presence of a parallel magnetic field. Also, discuss (4) the relation between energy gap and magnetic flux (ϕ) for armchair and zigzag systems by determining the lowest unoccupied state and the highest occupied state; and discuss (5) the magneto-optical selection rule under a parallel electric polarization.
9.An N y = 2 zigzag nanoribbon is the smallest 1D system. Investigate (1) the energy bands and (2) wave functions of the Hamiltonian matrix elements with nearest-neighboring hopping integrals of 2p z orbitals; and (3) the dependences on the magnetic flux through a hexagon.
10.Compare (1) the important differences between armchair (zigzag) nanotubes and zigzag (armchair) nanoribbons in terms of energy band, wave function, magnetic quantization and selection rule; and (2) propose concise physical pictures to explain them.
11.Discuss the evolution of magnetic quantization with dimension for AA, AB and ABC stackings by illustrating the sp2-bonding graphene-related systems.
