ABSTRACT
Graphene nanostructures have great potential for device applications. However, they can exhibit several counterintuitive electronic properties not present in ordinary semiconductor nanostructures. In this chapter, we review several of these graphene nanostructures. A rst example is a graphene antidot that possesses boundstates inside the antidot potential in the presence of a magnetic eld. As the range of the repulsive potential decreases in comparison to the magnetic length, the effective coupling constant between the potential and electrons becomes more repulsive, and then, it changes the sign and becomes attractive. This is a consequence of a subtle interplay between Klein tunneling and quantization of
Landau levels. In this regime, wavefunctions become anomalous with a narrow probability density peak inside the barrier and another broad peak outside the potential barrier with the width comparable to the magnetic length. The second example is a graphene parabolic dot in the presence of a magnetic eld. One counterintuitively nds that resonant quasibound states of both positive and negative energies exist in the energy spectrum. The presence of resonant quasi-bound states of negative energies is a unique property of massless Dirac fermions. Also, when the strength of the potential increases, resonant and nonresonant states transform into discrete anomalous states with a narrow probability density peak inside the well and another broad peak under the potential barrier with the width comparable to the magnetic length. The
Abstract ..................................................................................................................................................................................... 183 13.1 Part I: Gate Induced Antidots and Dots in a Magnetic Field .......................................................................................... 184
13.1.1 Antidot in Magnetic Fields .................................................................................................................................. 184 13.1.1.1 Effect of Magnetic Field: Eigenenergies and Eigenfunctions ............................................................... 185 13.1.1.2 Induced Density of Filled LLs .............................................................................................................. 185 13.1.1.3 Scaling Properties of Induced Density of Chiral and Nonchiral Dirac Fermions in Magnetic Fields ...... 187 13.1.1.4 Antidot in a Zigzag Ribbon in Magnetic Fields ................................................................................... 188 13.1.1.5 Periodic Antidots in Magnetic Fields ................................................................................................... 189
13.1.2 Parabolic Dot in Magnetic Fields ........................................................................................................................ 189 13.1.2.1 Coupling between Conduction and Valence Band States ..................................................................... 190 13.1.2.2 Resonant, Nonresonant, and Anomalous States ................................................................................... 192 13.1.2.3 Scaling and Optical Properties ............................................................................................................. 194
13.2 Part II: Dots with Zigzag and Armchair Edges in a Magnetic Field ............................................................................... 194 13.2.1 Rectagular Dot in Magnetic Fields ...................................................................................................................... 194
13.2.1.1 Zigzag Edges in B = 0 ........................................................................................................................... 195 13.2.1.2 Rectangular Dot with Two Zigzag Edges and Two Armchair Edges in B = 0 ..................................... 196 13.2.1.3 Rectangular Dot in Magnetic Fields ..................................................................................................... 197
13.3 Part III: Armchair Ribbons ............................................................................................................................................. 198 13.3.1 Spintronic Properties of One-Dimensional Electron Gas in Graphene Armchair Ribbons................................ 198
13.3.1.1 Formation of One-Dimensional Subbands in Armchair Ribbons ........................................................ 199 13.3.1.2 Exchange Self-Energy of Doped Armchair Ribbon ............................................................................. 200 13.3.1.3 Phase Diagram ...................................................................................................................................... 201
Appendix 13A: Derivation of the Dirac Equation from Tight-Binding Approach ................................................................... 201 Appendix 13B: Derivation of Effective Schrödinger Equation ................................................................................................ 204 Appendix 13C: Graphene LLs .................................................................................................................................................. 206 Acknowledgments ..................................................................................................................................................................... 207 References ................................................................................................................................................................................. 208
energy conduction subband of graphene armchair ribbons. Bulk magnetic properties of it may sensitively depend on the width of the ribbon. For ribbon widths Lx = 3Ma0, depending on the value of the Fermi energy, a ferromagnetic or paramagnetic state can be stable while for Lx = (3M + 1)a0, the paramagnetic state is stable (M is an integer and a0 is the length of the unit cell). Ferromagnetic and paramagnetic states are well suited for spintronic applications.
