石墨烯TE模表面等离子体波和表面等离子体波导的特性

万鹏; 杨翠红

doi:doi:10.3788/AOS201737.1124002

光学学报, 2017, 37 (11): 1124002, 网络出版: 2018-09-07

石墨烯TE模表面等离子体波和表面等离子体波导的特性下载： 1270次

Properties of Graphene TE Mode Surface Plasmons and Surface Plasmon Waveguides

论文大纲

万鹏杨翠红 ^*

作者单位

南京信息工程大学物理与光电工程学院, 江苏南京 210044

表面光学表面等离子体波石墨烯平板波导横电模 optics at surfaces surface plasmons graphene parallel-plated waveguide transverse electric mode

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摘要

基于含石墨烯的双/三层介质结构中的光学色散方程,研究了覆层和基底层材料对石墨烯表面等离子体波横电(TE)模的影响。计算结果表明,近红外波段内,石墨烯表面等离子体波TE模的性质对覆层和基底层介电常数的差值极其敏感。当覆层和基底层介电常数出现微小差异时,TE模可以进行传输。随着两介电常数差值的增大,TE模的有效折射率显著增大,甚至超过覆层折射率,而传播损耗不断减小。对于三层介质结构,相邻两介质分界面间添加单层石墨烯形成了平板波导结构,研究该结构发现,当传导层和基底层介电常数相近时,其夹层的石墨烯对波导TE模的调控尤其显著,这种调控来源于电磁波耦合到石墨烯形成的表面等离子体波TE模。这些结果为设计调制器、检测器和过滤器等石墨烯表面等离子体波导器件提供了理论支持。

Abstract

The influence of the cladding and substrate materials on graphene transverse electric (TE) mode surface plasmonic properties have been theoretically investigated based on the optical dispersion equation in the two-layer/three-layer dielectric system. It is found that the TE mode is highly sensitive to the dielectric contrast between the cladding and the substrate materials on both sides of graphene in the near-infrared regime. When the dielectric contrast between the cladding and the substrate is tiny, the TE mode surface plasmons can be transmitted. As the dielectric contrast increases, the TE mode dispersion curve behaves differently. The effective refractive index of the TE mode increases obviously and the propagation loss decreases continuously. For the three-layer parallel-plated waveguide structure with graphene embedded between two dielectric medium layers, the numerical results show that the control for the TE mode is significant especially when the dielectric constants for the propagation layer and the substrate layer is close to each other. It originates from the coupling between the electromagnetic wave and graphene leading to the surface plasmons between them. The results provide a theoretical support for the design of graphene surface plasmons based optoelectronic waveguides, such as modulators, detectors, and filters.

1 引言

表面等离子体波(SPs)是一种非辐射电磁表面波。SPs由电磁波耦合到金属表面,经电荷激发形成,并在介电常数相反的界面上传播^[1]。SPs局域性强,可以突破衍射极限,因而在光学生物传感器、发射器和相关波导器件^[2-3]等许多方面具有潜在的应用价值。一般情况下,采用贵金属支持SPs,但是通常所用的贵金属SPs传播损耗太大,并且在共振频率处发生饱和^[4],故需要寻求新材料来取代贵金属。

石墨烯^[5]是一种以六边形蜂窝状结构排列的二维单层碳原子结构。石墨烯的能带结构为线性,且其价带与导带相交于狄拉克点^[6]。这使得石墨烯具有许多优异的特性^[7],例如载流子迁移率高^[8],载流子响应快,静电调谐宽,近红外和可见光波段的光吸收率为2.3%^[9]等。有学者将石墨烯引入SPs,得到了石墨烯表面等离子体波(GSPs)。研究表明,GSPs约束性更强,传输距离更远,损耗更低,在纳米光电、等离子体领域表现出巨大的应用前景^[10]。通过近场红外纳米成像技术,Fei等^[11-12]已用实验证明了GSPs的存在。GSPs的激发则可以通过在原子力显微镜上施加红外光束^[11]、利用偶极子发射器^[13]或者设计石墨烯的几何结构^[14]等方式实现。

Falkovsky等^[15]给出了石墨烯的电导率为复数。电导率虚部的正负决定了石墨烯相对介电函数的正负。当电导率虚部为正数时,石墨烯表现为金属性,可以支持横磁(TM)模的SPs。当电导率虚部为负数时,石墨烯支持横电(TE)模的SPs^[16-17],这一性质与只支持TM模传播的金属不同。此外,GSPs比金属支持的SPs更具可控性,可以通过化学掺杂、施加电场、磁场以及栅极电压等方法调节GSPs的振幅和波长^[18-19]。

根据石墨烯的电导率特性,TM模的GSPs传播范围大约在太赫兹(THz)和远红外波段,而TE模的大约在近红外波段^[20]。很多文献报道了TM模特性的研究结果,如含石墨烯的双层结构^[21-23],三层介质的石墨烯平板波导结构^[24-27],多层介质添加单层石墨烯形成的结构^[28-29],介质加载型石墨烯结构^[3,30],高脊状、带状、锥状石墨烯结构^[2,22,31]等。相比于高损耗的贵金属SPs,这些结构中TM模的GSPs都表现出更强的约束性和更好的可控性^[32]。目前,有关约束性相对较弱的TE模的研究较少。但是TE模的GSPs损耗低、极化特性强、相速度快等特点在光电方面极具应用价值。Mikhailov等^[16]在2007年预测了当石墨烯电导率的虚部为负时,GSPs存在TE模。Jablan等^[33]在2011年预测了双层GSPs也存在TE模,且其约束性比单层GSPs强,同时认为双层GSPs具有很强的极化特性。Bao等^[34]利用波导在漏模处TE模损耗小于TM模的特性制成了基于石墨烯的光学偏振器。He等^[21,35]从理论上研究了夹杂石墨烯双层结构中TE模的GSPs,且在覆层和基底层介电常数出现较大差值时采用了增强基底材料的相对磁导率来实现TE模的GSPs,结果表明TE模的色散曲线出现了较大的改变。

本文通过控制覆层和基底层介电常数的微小差异同样实现了TE模GSPs的传输,并且TE模的色散曲线也出现了较大的改变。在此基础上,研究了石墨烯平板波导中TE模对介质的依赖性和石墨烯的调控作用。该研究可以为石墨烯表面等离子体器件中介质材料的选择提供理论基础,为制备低传播损耗的石墨烯表面等离子体器件提供了有益的建议,也有利于全面且准确地实现表面等离子体器件的有效调控。

2 理论模型

图1为含石墨烯的三层介质结构(石墨烯平板结构),相对介电常数分别为ε₁,ε₂,ε₃,相对磁导率分别为μ₁,μ₂,μ₃,传导层的厚度为d,覆层和基底层厚度为无穷大,在不同的介质之间分别添加一层石墨烯,形成a,b石墨烯分界面层。石墨烯层很薄,可以忽略其厚度的影响,分别采用电导率σ₁和σ₂描述界面电导。

图 1. 石墨烯平板结构

Fig. 1. Slab structure of graphene

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根据Kubo公式^[13],石墨烯的电导率σ_graphene源于带内电导率σ_intra和带间电导率σ_inter,σ_graphene=σ_intra+σ_inter。其中,带内电导率σ_intra为

\begin{matrix} σ_{intra} (ω) = \frac{2 i e^{2} K_{B} T}{π ћ (ω + i τ^{- 1})} \ln [2_{} \cos h (μ_{c} / 2 K_{B} T)], (1) \end{matrix}

带间电导率σ_inter近似表示为^[15]

\begin{matrix} σ_{inter} (ω) = \frac{e^{2}}{4 ћ} [θ (ω - 2 μ_{c}) - \frac{i}{2 π} \ln \frac{(ω + 2 μ_{c})^{2}}{(ω - 2 μ_{c})^{2} + (2 K_{B} {T)}^{2}}], (2) \end{matrix}

式中ω为频率,e为电子电荷,K_B为玻尔兹曼常数,T为温度,取T=300 K,μ_c为化学势,ћ为约化普朗克常数,τ为弛豫时间,取τ=5×10^-13 s,θ(ω-2μ_c)为阶跃函数。

对于平行波导,y方向宽度无穷大,不受限。覆层和基底层介质在x方向无穷大。以下计算过程中传播因子均使用exp(-iωt),对于TE模,电场仅存在y方向的分量,各区间的电场分布为

\begin{matrix} E_{y} (x) = \{\begin{matrix} A_{} \exp (- i γ_{1} x), & x > 0 \\ B_{} \sin γ_{2} x + co s_{} γ_{2} x, & - d < x < 0 \\ C_{} \exp (i γ_{3} x), & x < - d \end{matrix}, (3) \end{matrix}

式中 $\begin{matrix} γ_{j}^{2} \end{matrix}$ =ε_jμ_j $\begin{matrix} k_{0}^{2} \end{matrix}$ -β²(j=1,2,3),ε_j,μ_j分别为各个介质层的相对介电常数和相对磁导率,k₀为真空波数,β为传播常数,大多情况下,μ_j=1,则ε_j= $\begin{matrix} n_{j}^{2} \end{matrix}$ 。

根据麦克斯韦方程得H_z= $\begin{matrix} \frac{- i}{ω μ_{j} μ_{0}} \end{matrix} \begin{matrix} \frac{\partial E_{y}}{\partial x} \end{matrix}$ ,根据边界条件得:

当x=0时,

\begin{matrix} \{\begin{matrix} E_{y}^{1} = E_{y}^{2} \\ H_{z}^{1} - H_{z}^{2} = - σ_{1} E_{y} \end{matrix}; (4) \end{matrix}

当x=-d时,

\begin{matrix} \{\begin{matrix} E_{y}^{2} = E_{y}^{3} \\ H_{z}^{2} - H_{z}^{3} = - σ_{2} E_{y} \end{matrix} 。 (5) \end{matrix}

综合(3)~(5)式可得色散方程为

\begin{matrix} \tan (γ_{2} d) = i \frac{C_{1} + C_{2}}{C_{1} C_{2} + 1}, (6) \end{matrix}

式中C₁= $\begin{matrix} (1 - \frac{σ_{1} ω μ_{0} μ_{1}}{γ_{1}}) \end{matrix} \begin{matrix} {(\frac{μ_{1} γ_{2}}{μ_{2} γ_{1}})}^{- 1} \end{matrix}$ ,C₂= $\begin{matrix} (1 - \frac{σ_{2} ω μ_{0} μ_{3}}{γ_{3}}) \end{matrix} \begin{matrix} {(\frac{μ_{3} γ_{2}}{μ_{2} γ_{3}})}^{- 1} \end{matrix}$ 。

对于TM模,(6)式的色散方程形式不变,此时C₁= $\begin{matrix} (\frac{ε_{2} γ_{1}}{ε_{1} γ_{2}}) \end{matrix} \begin{matrix} {(1 - \frac{σ_{1} γ_{1}}{ω ε_{1} ε_{0}})}^{- 1} \end{matrix}$ ,C₂= $\begin{matrix} (\frac{ε_{2} γ_{3}}{ε_{3} γ_{2}}) \end{matrix} \begin{matrix} {(1 - \frac{σ_{2} γ_{3}}{ω ε_{3} ε_{0}})}^{- 1} \end{matrix}$ 。

对于(6)式,令ε₂=ε₃,σ₂=0,d无穷大,即退化得到双层介质中间夹一层石墨烯的TE、TM方程分别为^[21]

\begin{matrix} \begin{matrix} \frac{γ_{1}}{μ_{1}} + \frac{γ_{2}}{μ_{2}} - iσω μ_{0} = 0, (7) \\ \frac{ε_{1}}{γ_{1}} + \frac{ε_{2}}{γ_{2}} + \frac{iσ}{ω e_{0}} = 0, (8) \end{matrix} \end{matrix}

式中γ_j=k₀ $\begin{matrix} \sqrt[]{β^{2} - ε_{j} μ_{j}} \end{matrix}$ ,j=1,2。

3 讨论

通过施加栅极电压、化学掺杂等方法可改变石墨烯的化学势μ_c,进一步实现对石墨烯电导率σ的调节。根据石墨烯的电导率公式[(1)式和(2)式],图2给出了近红外波段范围内石墨烯电导率的实部和虚部在不同化学势下随频率的变化关系。

在250~400 THz范围内,电导率实部随频率的增加不断增加,最终达到σ_min[σ_min=πe²/(2h)]时饱合。电导率虚部始终为负,满足TE模GSPs的传输条件。不同化学势下电导率虚部在250 THz处差别较为明显,而在400 THz处,各化学势下的电导率都趋向于零,差别随频率的增大而减小。

图 2. 不同化学势下石墨烯的电导率随频率的变化。(a)实部;(b)虚部

Fig. 2. Change of graphene conductivity with frequency at different chemical potentials. (a) Real part; (b) imaginary part

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首先分析含石墨烯双层介质中TE模的传播特性,令ε₂=ε₃,σ₂=0,d无穷大,则TE模的色散方程为(7)式。对于悬空的石墨烯,ε₁=ε₂=1,TE模随石墨烯化学势变化的传播特性如图3所示,其中有效折射率n_eff=Re(β/k₀)为归一化传播常数的实部,Im(β/k₀)为归一化传播常数的虚部。从图3可以看出,同一频率下,随着化学势的增加,TE模中有效折射率n_eff不断上升,说明石墨烯光电导能够对TE模进行调控。但n_eff始终小于空气的折射率,说明TE模较弱。虚部Im(β/k₀)也发生变化,说明石墨烯光电导也能对TE模损耗进行调控。

图 3. 不同化学势下悬空的石墨烯结构中TE模色散随频率的变化。(a)有效折射率neff;(b) Im(β/k0)

Fig. 3. Dispersion of TE mode in a suspended graphene structure varies with frequency at different chemical potentials. (a) Effective refractive index neff; (b) Im(β/k0)

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为了更好地阐述覆层和基底层介电常数出现微小差异时对TE模GSPs的影响,首先考虑石墨烯覆层和基底介质同为Si、SiO₂、SiC等应用广泛的介质材料的情况,即ε₁=ε₂,如图4所示。随着覆层和基底介质的折射率同时不断增加,TE模中n_eff也不断增加,但都略小于覆层和基底的介质折射率,表明TE模的GSPs弱于TM模的。随着覆层和基底介质的折射率同时不断增加,TE模中Im(β/k₀)却在不断减小,意味着TE模的GSPs在非空气介质中比在空气中衰减更小、传播更远,更有利于实际应用。对比文献[ 20],TM模的GSPs损耗远大于TE模,并且随着介质折射率的增加,这种损耗在不断增加。

图 4. ε1=ε2时双层介质结构中TE模色散随频率的变化(Si、SiC、SiO2介质的折射率分别为3.67,2.57,1.53;μc=0.5 eV)。(a) neff; (b) Im(β/k0)

Fig. 4. Dispersion of TE mode varies with frequency in a two-layer dielectric structure as ε1=ε2 (the refractive indices of Si, SiC and SiO2 are 3.67, 2.57 and 1.53, respectively; μc=0.5 eV). (a) neff; (b) Im(β/k0)

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当ε₁和ε₂差异较小,即ε₁≈ε₂时,TE模的GSPs仍然可以传播。如图5所示,选取覆层和基底材料都为SiO₂,但是介电常数有微小差别,ε₁(SiO₂)≠ε₂(SiO₂),其中,覆层介电常数ε₁(SiO₂)= $\begin{matrix} n_{Si O_{2}}^{2} \end{matrix}$ =2.3409。

当基底介电常数ε₂(SiO₂)不断增大,但增大量很小时,TE模中n_eff不断增大,甚至可以超过覆层SiO₂的折射率n₁= $\begin{matrix} \sqrt[]{ε_{1}} \end{matrix}$ =1.53,而在覆层和基底的介电常数相同时,TE模中n_eff始终小于覆层折射率。可见,覆层和基底材料介电常数的微小差异能够明显改变TE模的色散曲线,并且随着这种差距的增大,传播损耗降低。随着ε₂的继续增大,研究发现不能继续获得TE模,但可以通过对基底材料相对磁导率μ₂的增强来实现TE模的传播^[21,35]。这种方式需要相当大的磁导率,因此,研究内容进一步补充了如何实现TE模GSPs。

图 5. ε1≈ε2时TE模色散随频率的变化[ε1(SiO2)=2.3409;ε2(SiO2)分别为2.3409,2.3410,2.3411,2.3412;μc=0.5 eV]。(a) neff; (b) Im(β/k0)

Fig. 5. Dispersion of TE mode varies with frequency as ε1≈ε2 [ε1(SiO2)=2.3409; ε2(SiO2) are 2.3409, 2.3410, 2.3411, and 2.3412, respectively; μc=0.5 eV]. (a) neff; (b) Im(β/k0)

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研究三层介质石墨烯平板波导TE模的传播特性,仅考虑基模传播的特性。对于(6)式,令覆层为空气,即n₁=1;传导层介质为SiO₂,折射率n₂=1.468,传导层厚度d=7 μm。

当n₃=1.460时,平板波导形成了不对称结构。为了探讨a,b层分别添加石墨烯对波导传输TE模的影响,对不添加任何石墨烯的波导、a层添加石墨烯的波导、b层添加石墨烯的波导、a层和b层同时添加石墨烯的波导的TE模色散曲线进行对比,如图6所示。

图 6. 波导中a层和b层分界面有无添加石墨烯时TE模色散随频率的变化(n3=1.460,μc=0.5 eV)。(a) neff; (b) Im(β/k0)

Fig. 6. Dispersion of TE mode varies with frequency in the waveguide with and without graphene in the interfaces a and b (n3=1.460, μc=0.5 eV). (a) neff; (b) Im(β/k0)

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图6(a)表明,对于不存在石墨烯的波导,TE模中n_eff随频率不断增加,但小于传导层的折射率n₂=1.468,添加石墨烯后对波导TE模色散关系有了显著的调制。这种调制的机理是波导中TE模的电磁波耦合到石墨烯表面形成了TE模的GSPs,这一点与文献[ 26]中所述一致。由于波导结构的不对称性,a层和b层分别添加石墨烯对波导TE模的调控能力并不相同。绿色实线和蓝色虚线几乎重叠,表明b层添加石墨烯与a层和b层同时添加石墨烯的TE模色散曲线几乎一致。这说明b层石墨烯的调控作用几乎掩盖了a层石墨烯的调控作用,意味着电磁波耦合到b层石墨烯表面形成的TE模GSPs明显强于a层石墨烯。这与上文所述双层结构中覆层和基底层介电常数相近时TE模GSPs容易形成的情况相吻合。从图6(b)可以看出,这种很强的调控伴随着更大的损耗。

研究石墨烯调控TE模对基底材料的依赖关系,主要考虑b层是否添加石墨烯与基底材料折射率两个方面,如图7所示。从图7(a)看出,随着基底折射率与传导层折射率差值越来越大,b层添加石墨烯对波导TE模的调控越来越弱,说明TE模的电磁波耦合到b层石墨烯形成的TE模GSPs越来越少,这也印证了上文双层结构的结论。从7(b)图可以看出,添加b层石墨烯增加了波导的损耗,并且b层石墨烯调控作用越强,损耗越明显。

图 7. 基底介质波导中b层分界面有无添加石墨烯且n3不同时TE模色散随频率的变化(μc=0.5 eV)。 (a) neff; (b) Im(β/k0)

Fig. 7. Dispersion of TE mode varies with frequency for different dielectric constant n3 in substrate waveguide with and without graphene in the interface b (μc=0.5 eV). (a) neff; (b) Im(β/k0)

下载图片查看所有图片

利用图8分析了b层石墨烯电导率对波导TE模的调控。其中,μ_c(a), μ_c(b)分别为a,b层添加石墨烯的化学势。从图8(a)可以看出,在250 THz附近,石墨烯电导率对TE模的调控显著增强,这与图2(b)中石墨烯电导率虚部在250 THz附近变化明显有着密不可分的联系。对调控b层石墨烯电导率对损耗的影响并不明显。总而言之,调节石墨烯的电导率可以更加有效地调控平板波导的传输。

图 8. b层分界面中石墨烯化学势不同时TE模色散随频率的变化[μc(a)=0.2 eV,n3=1.460]。 (a) neff; (b) Im(β/k0)

Fig. 8. Dispersion of TE mode varies with frequency at different chemical potentials of graphene in the interface b [μc(a)=0.2 eV, n3=1.460]. (a) neff; (b) Im(β/k0)

下载图片查看所有图片

4 结论

研究了近红外波段含石墨烯双层介质间的TE模GSPs以及三层介质石墨烯平板波导中TE模的传输特性。结果表明,当覆层和基底层材料介电常数相同时,TE模的GSPs可以传输。当覆层和基底层材料介电常数出现微小的差异时,TE模的GSPs依旧可以传输,但TE模的色散曲线出现了较大的改变,有效折射率甚至可以超过覆层的折射率。此外,三层介质石墨烯平板波导中,TE模的调控也有赖于介质材料。当传导层和基底层材料介电常数相近时,夹层的石墨烯光电导对波导TE模调控尤其显著。结果还表明,TE模的石墨烯表面等离子体波和石墨烯平板波导中的TE模传输特性对介质材料存在依赖关系。这有助于对石墨烯表面等离子体相关光电材料应用的理论分析和认识。

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万鹏, 杨翠红. 石墨烯TE模表面等离子体波和表面等离子体波导的特性[J]. 光学学报, 2017, 37(11): 1124002. Peng Wan, Cuihong Yang. Properties of Graphene TE Mode Surface Plasmons and Surface Plasmon Waveguides[J]. Acta Optica Sinica, 2017, 37(11): 1124002.

石墨烯TE模表面等离子体波和表面等离子体波导的特性下载： 1270次

1 引言

2 理论模型

图 1. 石墨烯平板结构

Fig. 1. Slab structure of graphene

3 讨论

图 2. 不同化学势下石墨烯的电导率随频率的变化。(a)实部;(b)虚部

Fig. 2. Change of graphene conductivity with frequency at different chemical potentials. (a) Real part; (b) imaginary part

图 3. 不同化学势下悬空的石墨烯结构中TE模色散随频率的变化。(a)有效折射率neff;(b) Im(β/k0)

Fig. 3. Dispersion of TE mode in a suspended graphene structure varies with frequency at different chemical potentials. (a) Effective refractive index neff; (b) Im(β/k0)

图 4. ε1=ε2时双层介质结构中TE模色散随频率的变化(Si、SiC、SiO2介质的折射率分别为3.67,2.57,1.53;μc=0.5 eV)。(a) neff; (b) Im(β/k0)

Fig. 4. Dispersion of TE mode varies with frequency in a two-layer dielectric structure as ε1=ε2 (the refractive indices of Si, SiC and SiO2 are 3.67, 2.57 and 1.53, respectively; μc=0.5 eV). (a) neff; (b) Im(β/k0)

图 5. ε1≈ε2时TE模色散随频率的变化[ε1(SiO2)=2.3409;ε2(SiO2)分别为2.3409,2.3410,2.3411,2.3412;μc=0.5 eV]。(a) neff; (b) Im(β/k0)

Fig. 5. Dispersion of TE mode varies with frequency as ε1≈ε2 [ε1(SiO2)=2.3409; ε2(SiO2) are 2.3409, 2.3410, 2.3411, and 2.3412, respectively; μc=0.5 eV]. (a) neff; (b) Im(β/k0)

图 6. 波导中a层和b层分界面有无添加石墨烯时TE模色散随频率的变化(n3=1.460,μc=0.5 eV)。(a) neff; (b) Im(β/k0)

Fig. 6. Dispersion of TE mode varies with frequency in the waveguide with and without graphene in the interfaces a and b (n3=1.460, μc=0.5 eV). (a) neff; (b) Im(β/k0)

图 7. 基底介质波导中b层分界面有无添加石墨烯且n3不同时TE模色散随频率的变化(μc=0.5 eV)。 (a) neff; (b) Im(β/k0)

Fig. 7. Dispersion of TE mode varies with frequency for different dielectric constant n3 in substrate waveguide with and without graphene in the interface b (μc=0.5 eV). (a) neff; (b) Im(β/k0)

图 8. b层分界面中石墨烯化学势不同时TE模色散随频率的变化[μc(a)=0.2 eV,n3=1.460]。 (a) neff; (b) Im(β/k0)

Fig. 8. Dispersion of TE mode varies with frequency at different chemical potentials of graphene in the interface b [μc(a)=0.2 eV, n3=1.460]. (a) neff; (b) Im(β/k0)

4 结论

Article Outline

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石墨烯TE模表面等离子体波和表面等离子体波导的特性 下载： 1270次

1 引言

2 理论模型

图 1. 石墨烯平板结构

Fig. 1. Slab structure of graphene

3 讨论

图 2. 不同化学势下石墨烯的电导率随频率的变化。(a)实部;(b)虚部

Fig. 2. Change of graphene conductivity with frequency at different chemical potentials. (a) Real part; (b) imaginary part

图 3. 不同化学势下悬空的石墨烯结构中TE模色散随频率的变化。(a)有效折射率neff;(b) Im(β/k0)

Fig. 3. Dispersion of TE mode in a suspended graphene structure varies with frequency at different chemical potentials. (a) Effective refractive index neff; (b) Im(β/k0)

图 4. ε1=ε2时双层介质结构中TE模色散随频率的变化(Si、SiC、SiO2介质的折射率分别为3.67,2.57,1.53;μc=0.5 eV)。(a) neff; (b) Im(β/k0)

Fig. 4. Dispersion of TE mode varies with frequency in a two-layer dielectric structure as ε1=ε2 (the refractive indices of Si, SiC and SiO2 are 3.67, 2.57 and 1.53, respectively; μc=0.5 eV). (a) neff; (b) Im(β/k0)

图 5. ε1≈ε2时TE模色散随频率的变化[ε1(SiO2)=2.3409;ε2(SiO2)分别为2.3409,2.3410,2.3411,2.3412;μc=0.5 eV]。(a) neff; (b) Im(β/k0)

Fig. 5. Dispersion of TE mode varies with frequency as ε1≈ε2 [ε1(SiO2)=2.3409; ε2(SiO2) are 2.3409, 2.3410, 2.3411, and 2.3412, respectively; μc=0.5 eV]. (a) neff; (b) Im(β/k0)

图 6. 波导中a层和b层分界面有无添加石墨烯时TE模色散随频率的变化(n3=1.460,μc=0.5 eV)。(a) neff; (b) Im(β/k0)

Fig. 6. Dispersion of TE mode varies with frequency in the waveguide with and without graphene in the interfaces a and b (n3=1.460, μc=0.5 eV). (a) neff; (b) Im(β/k0)

图 7. 基底介质波导中b层分界面有无添加石墨烯且n3不同时TE模色散随频率的变化(μc=0.5 eV)。 (a) neff; (b) Im(β/k0)

Fig. 7. Dispersion of TE mode varies with frequency for different dielectric constant n3 in substrate waveguide with and without graphene in the interface b (μc=0.5 eV). (a) neff; (b) Im(β/k0)

图 8. b层分界面中石墨烯化学势不同时TE模色散随频率的变化[μc(a)=0.2 eV,n3=1.460]。 (a) neff; (b) Im(β/k0)

Fig. 8. Dispersion of TE mode varies with frequency at different chemical potentials of graphene in the interface b [μc(a)=0.2 eV, n3=1.460]. (a) neff; (b) Im(β/k0)

4 结论

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石墨烯TE模表面等离子体波和表面等离子体波导的特性下载： 1270次