We present the analytical solution of the wavefunction and energy dispersion of armchair graphene nanoribbons (GNRs) based on the tight-binding approximation. By imposing hard-wall boundary condition, we find that the wavevector in the confined direction is discretized. This discrete wavevector serves as the index of different subbands. Our analytical solutions of wavefunction and associated energy dispersion reproduce the numerical tight-binding results and the solutions based on the k · p approximation. In addition, we also find that all armchair GNRs with edge deformation have energy gaps, which agrees with recently reported first-principles calculations.

Stimulated by recent advances in isolating graphene, we discovered that quantum dot can be trapped in Z-shaped graphene nanoribbon junciton. The topological structure of the junction can confine electronic states completely. By varying junction length, we can alter the spatial confinement and the number of discrete levels within the junction. In addition, quantum dot can be realized regardless of substrate induced static disorder or irregular edges of the junction. This device can be used to easily design quantum dot devices. This platform can also be used to design zero-dimensional functional nanoscale electronic devices using graphene ribbons.

Rectangular graphene nanodots have both armchair and zigzag edges, which can be terminated with a variety of atoms or molecular groups. Our first-principles study shows that edge chemical modification can alter the electronic structure of graphene nanodots significantly. We find that when saturated with different atoms or molecular groups ͑such as -H, -F, -OH, etc.͒ on the zigzag edges, the graphene nanodots show a spin-polarized ground state, but with magnetic moment, spin density and energy gap are strongly dependent on the type of saturating atom or molecular group. Our results indicate that graphene nanodots have great potential to serve as future molecular sensor and transistor devices. I. I NTRODUCTIONWith experimental advances in the fabrication 1-5 of graphene materials, two-dimensional ͑2D͒ graphene, onedimensional ͑1D͒ graphene nanoribbons ͑GNRs͒ and zerodimensional ͑0D͒ graphene nanodots have been extensively studied from both experimental and theoretical points of view. 6-13 The relativistic characteristics 6,7 and outstanding electronic properties involving ballistic transport and large coherence length 2 make graphene an excellent platform for studying quantum electrodynamics ͑QED͒ phenomena. 8 It is also a promising candidate material for future nanoelectronic 3,9-11 and nanospintronic 12,13 applications. What is more exciting is the very recent report of the successful fabrication of graphene quantum dots as small as 20 nm, with which a purely graphene-based single-electron transistor was realized and studied. 14 Two particular types of GNRs with armchair and zigzag shaped edges have been extensively studied. [15][16][17][18][19] In the tightbinding approximation, armchair graphene nanoribbons ͑A-GNRs͒ are predicted to be metallic when n =3m +2, where n is the width and m is an integer, and are insulating otherwise, 11,15-21 while first-principles studies have shown that all A-GNRs have opened gaps with the introduction of edge deformations. 11 Zigzag graphene nanoribbons ͑Z-GNRs͒ are semiconducting with nonzero energy gaps due to the existence of ferromagnetically ordered edge states at each zigzag edge and an antiferromagnetic arrangement of spins between two zigzag edges. 11,12 The application of an external electrical field across the ribbon can drive Z-GNRs into a half-metallic state where metallic electrons with one spin orientation coexist with insulating electrons having the other spin orientation. 12,22 Following the investigations of quasi-one-dimensional graphene nanoribbons, much research work has focused on the electronic and magnetic properties of quasi-zerodimensional graphene nanodots. [22][23][24][25][26][27] Due to the existence of both armchair and zigzag edges, finite 0D graphene nanodots also adopt spin-polarized edge states similar to 1D zigzag graphene nanoribbons. The half-metallic nature of 1D zigzag graphene nanoribbons under an external transverse electrical field is predicted to be preserved for 0D graphene nanodots. 24 However, quantum confinement and finite-size effec...

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In this paper, we focus on a graphene heterojunction device: a Z-shaped graphene nanoribbon, which consists of two armchair leads and a zigzag junction. Based on the Landauer-Büttiker formula and the tight binding model, we found that the rectifying behavior can be achieved by applying an external gate voltage in the heterjunction region. We also found that the rectification effect is independent of junction width and length, it is an intrinsic property of the Z-junction graphene nanoribbon. This platform can be used to design and study functional graphene nanoscale devices.