# Guided Modes in a Double-Well Asymmetric Potential of a Graphene Waveguide

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## Abstract

**:**

## 1. Introduction

## 2. Guided Mode and Dispersion Equation for a Double-Well Potential

## 3. Characteristics of the Guided Modes in a Double-Well Potential

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of (

**a**) a double-well potential on graphene; (

**b**) guided modes in the region $(0,{h}_{1})$; (

**c**) guided modes in the region $(0,{h}_{1}+{h}_{2})$.

**Figure 2.**The critical angle ${\theta}_{ij}$ as a function of the incidence energy E. The physical parameters are: ${V}_{1}={V}_{4}=100$ meV, ${V}_{2}=0$ meV, ${V}_{3}=60$ meV, ${h}_{1}=100$ nm, ${h}_{2}=80$ nm.

**Figure 3.**Graphical determination of ${k}_{2x}{h}_{1}$ for guided modes in $(0,{h}_{1})$. The

**red**and

**blue**curves correspond to $tan({k}_{2x}{h}_{1})$ and $F({k}_{2x}{h}_{1})$, respectively. The energy of incident electron is (

**a**) $E=56$ meV; (

**b**) $E=94$ meV; and (

**c**) $E=110$ meV. The other parameters are the same as those in Figure 2.

**Figure 4.**(

**a**) the wave function of guided modes as a function of distance corresponding to the intersection $({k}_{2x}{h}_{1}=2.9257,\theta ={69.8694}^{\mathrm{o}})$ in Figure 3a. The solid curve and the dashed lines corresponds to ${\mathsf{\Psi}}_{A}$ and $-i{\mathsf{\Psi}}_{B}$, respectively. The vertical lines represent the boundaries of the waveguide; and (

**b**) the corresponding probability current density within the graphene waveguide for the guided mode. The solid black lines represent boundaries of the waveguide.

**Figure 5.**The wave function ${\mathsf{\Psi}}_{A}$ (solid) and $-i{\mathsf{\Psi}}_{B}$ (dashed) of guided modes as a function of distance corresponding to the intersections in Figure 3b: (

**a**) ${k}_{2x}{h}_{1}=2.8959$, $\theta ={78.2908}^{\mathrm{o}}$; (

**b**) ${k}_{2x}{h}_{1}=5.7775$, $\theta ={66.1160}^{\mathrm{o}}$; (

**c**) ${k}_{2x}{h}_{1}=8.6220$, $\theta ={52.8267}^{\mathrm{o}}$; and (

**d**) ${k}_{2x}{h}_{1}=11.3661$, $\theta ={37.1996}^{\mathrm{o}}$.

**Figure 6.**The wave function ${\mathsf{\Psi}}_{A}$ (solid) and $-i{\mathsf{\Psi}}_{B}$ (dashed) of guided modes as a function of distance corresponding to the intersections in Figure 3c: (

**a**) ${k}_{2x}{h}_{1}=2.9025$, $\theta ={79.9900}^{\mathrm{o}}$; (

**b**) ${k}_{2x}{h}_{1}=5.7934$, $\theta ={69.6994}^{\mathrm{o}}$; (

**c**) ${k}_{2x}{h}_{1}=8.6556$, $\theta ={58.7782}^{\mathrm{o}}$; (

**d**) ${k}_{2x}{h}_{1}=11.4531$, $\theta ={46.6948}^{\mathrm{o}}$; and (

**e**) ${k}_{2x}{h}_{1}=14.0545$, $\theta ={32.6828}^{\mathrm{o}}$.

**Figure 7.**Graphical determination of ${k}_{2x}{h}_{1}$ for guided modes in $(0,{h}_{1}+{h}_{2})$. The

**red**and

**blue**curves correspond to $tan({k}_{2x}{h}_{1})$ and $F({k}_{2x}{h}_{1})$, respectively. The energy of incident electron is (

**a**) $E=94$ meV and (

**b**) $E=110$ meV. The other parameters are the same as those in Figure 2.

**Figure 8.**The wave function ${\mathsf{\Psi}}_{A}$ (solid) and $-i{\mathsf{\Psi}}_{B}$ (dashed) of guided modes as a function of distance corresponding to the intersections in Figure 7a: (

**a**) ${k}_{2x}{h}_{1}=13.4848$, $\theta ={19.0890}^{\mathrm{o}}$; (

**b**) ${k}_{2x}{h}_{1}=14.0965$, $\theta ={8.9296}^{\mathrm{o}}$; and (

**c**) ${k}_{2x}{h}_{1}=14.2552$, $\theta ={4.5134}^{\mathrm{o}}$.

**Figure 9.**The wave function ${\mathsf{\Psi}}_{A}$ (solid) and $-i{\mathsf{\Psi}}_{B}$ (dashed) of guided modes as a function of distance corresponding to the intersections in Figure 7b: (

**a**) ${k}_{2x}{h}_{1}=15.2675$, $\theta ={23.8914}^{\mathrm{o}}$; (

**b**) ${k}_{2x}{h}_{1}=16.0201$, $\theta ={16.3885}^{\mathrm{o}}$; and (

**c**) ${k}_{2x}{h}_{1}=16.6082$, $\theta ={5.9544}^{\mathrm{o}}$.

**Figure 10.**Probability current density within the graphene waveguide for the guided modes: (

**a**) $E=94$ meV, ${k}_{2x}{h}_{1}=5.7775$; (

**b**) $E=94$ meV, ${k}_{2x}{h}_{1}=14.0965$; (

**c**) $E=110$ meV, ${k}_{2x}{h}_{1}=5.7934$; and (

**d**) $E=110$ meV, ${k}_{2x}{h}_{1}=16.0201$.

**Figure 11.**Graphical determination of $tan({k}_{2x}{h}_{1})$ and $F({k}_{2x}{h}_{1})$ for guided modes in (

**a**) $(0,{h}_{1})$ and (

**b**) $(0,{h}_{1}+{h}_{2})$ with different values of ${h}_{2}$. Here, the incident energy of electrons is fixed at $E=94$ meV. The other parameters are the same as those in Figure 2.

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**MDPI and ACS Style**

Xu, Y.; Ang, L.K.
Guided Modes in a Double-Well Asymmetric Potential of a Graphene Waveguide. *Electronics* **2016**, *5*, 87.
https://doi.org/10.3390/electronics5040087

**AMA Style**

Xu Y, Ang LK.
Guided Modes in a Double-Well Asymmetric Potential of a Graphene Waveguide. *Electronics*. 2016; 5(4):87.
https://doi.org/10.3390/electronics5040087

**Chicago/Turabian Style**

Xu, Yi, and Lay Kee Ang.
2016. "Guided Modes in a Double-Well Asymmetric Potential of a Graphene Waveguide" *Electronics* 5, no. 4: 87.
https://doi.org/10.3390/electronics5040087