^{1,a)}, Ze Cheng

^{1}, Qi-Jun Zeng

^{1}, Jun-Pei Zhang

^{1}and Jian-Jun Zhang

^{1}

### Abstract

Motivated by the realization of the Dirac point with tunable Fermi velocity in low-dimensional systems, we investigate the guided modes in graphene waveguides corresponding to the electron motion (or the hole motion) in a symmetric velocity barrier. We find that the fundamental mode always exists, but the higher-order mode may disappear. These discrete guided modes imply that there is a lowest cutoff frequency for an incident electron and that the incident electrons with different angles may have different minimum cutoff frequencies. These interesting features will be helpful for the investigation on an electronic fiber.

This work was supported by the National Natural Science Foundation of China under Grant No. 10174024 and No. 10474025.

I. INTRODUCTION

II. MODEL AND METHOD

A. Model Hamiltonian

III. RESULTS AND DISCUSSIONS

IV. CONCLUSION

### Key Topics

- Graphene
- 21.0
- Dirac equation
- 18.0
- Bound states
- 8.0
- Evanescent waves
- 6.0
- Localized states
- 5.0

## Figures

(Color online) Schematic diagram of the graphene waveguides. (a) Three-dimensional schematic illustration of the device; a monolayer graphene sheet is on top of a silicon sheet, separated from it by a thick layer. The silicon is doped and connected to the electrode through a thin layer of silicon defined by selective etching. The interaction between electrons can be screened by a grounded metallic plate in the doped sheet, which can induce a renormalized Fermi velocity. (b) The model of velocity barriers: The lower panel describes the velocity barrier profile. In the left (or right) region, Fermi velocity can be controlled to be greater or less than that of graphene (the middle sheet). The spectrum of electron and hole are linear, and the tunable Fermi velocity is indicated by the slope of the linear spectrum. The cross points represent the Dirac points. (c) A cross-section of doped graphene sheet in (a).

(Color online) Schematic diagram of the graphene waveguides. (a) Three-dimensional schematic illustration of the device; a monolayer graphene sheet is on top of a silicon sheet, separated from it by a thick layer. The silicon is doped and connected to the electrode through a thin layer of silicon defined by selective etching. The interaction between electrons can be screened by a grounded metallic plate in the doped sheet, which can induce a renormalized Fermi velocity. (b) The model of velocity barriers: The lower panel describes the velocity barrier profile. In the left (or right) region, Fermi velocity can be controlled to be greater or less than that of graphene (the middle sheet). The spectrum of electron and hole are linear, and the tunable Fermi velocity is indicated by the slope of the linear spectrum. The cross points represent the Dirac points. (c) A cross-section of doped graphene sheet in (a).

(Color online) The phase diagram in momentum space. The dashed circle indicates the wavevector in the incident region I, and the solid circle indicates the wavevector in the transmitted region II for the velocity ratio and , respectively.

(Color online) The phase diagram in momentum space. The dashed circle indicates the wavevector in the incident region I, and the solid circle indicates the wavevector in the transmitted region II for the velocity ratio and , respectively.

(Color online) Graphical determination of for oscillating guided modes. The intersections show the existence of the guided modes. The solid and the dashed curves correspond to the and , respectively, where physical parameters are chosen to be nm for different cases: (a) meV, , (b) meV, , (c) meV, , and (d) meV, .

(Color online) Graphical determination of for oscillating guided modes. The intersections show the existence of the guided modes. The solid and the dashed curves correspond to the and , respectively, where physical parameters are chosen to be nm for different cases: (a) meV, , (b) meV, , (c) meV, , and (d) meV, .

(Color online) Energy spectrum of the bound states as a function of the angle of incident electron for different velocity ratio , where the physical parameter is chosen to be nm. The solid curve and the dashed curve correspond to and , respectively. The vertical solid curve and the vertical dashed curve correspond to TIR () for and TIR () for , respectively. The two horizontal dot dashed curve correspond to the case of Figs. 3(a) and 3(c) and the case of Figs. 3(b) and 3(d), respectively. n denotes the guided modes. The solid curve (*n* = 1) and the dashed curve (*n* = 1) denote the fundamental mode for and , respectively.

(Color online) Energy spectrum of the bound states as a function of the angle of incident electron for different velocity ratio , where the physical parameter is chosen to be nm. The solid curve and the dashed curve correspond to and , respectively. The vertical solid curve and the vertical dashed curve correspond to TIR () for and TIR () for , respectively. The two horizontal dot dashed curve correspond to the case of Figs. 3(a) and 3(c) and the case of Figs. 3(b) and 3(d), respectively. n denotes the guided modes. The solid curve (*n* = 1) and the dashed curve (*n* = 1) denote the fundamental mode for and , respectively.

(Color online) The wave function of guided modes as a function of the distance of graphene waveguide corresponding to the intersections in Fig. 3(a). The solid curve and the dashed curve correspond to and , respectively. The physical parameters are nm, , and meV for four guided modes: (a) and , (b) and , (c) and , and (d) and .

(Color online) The wave function of guided modes as a function of the distance of graphene waveguide corresponding to the intersections in Fig. 3(a). The solid curve and the dashed curve correspond to and , respectively. The physical parameters are nm, , and meV for four guided modes: (a) and , (b) and , (c) and , and (d) and .

(Color online) Probability current density distribution of electron states corresponding to graphene-guided modes. The physical parameters are identical to those in Fig. 5.

(Color online) Probability current density distribution of electron states corresponding to graphene-guided modes. The physical parameters are identical to those in Fig. 5.

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