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Graphene plasmons are important collective excitations in graphene, which play a key role in determining the optical properties of graphene. They have quite lots of unique features in comparison with classical plasmons in noble metals. Of them, the active tunability is the most attractive, which is realized by external gating (equivalently electric field). As is well known, graphene also has strong magnetic response (e.g. room temperature quantum Hall effect), so magnetic field can act as another degree of freedom for actively tuning graphene plasmons, with the new quasi particles being so-called graphene magneto-plasmons. Because of the two-dimensional nature of graphene, the numerical studies (or full wave simulations) of graphene magneto-plasmons are usually carried out through a three-dimensional approximation, e.g. treating two-dimensional graphene as a very thin three-dimensional film. Actually, this treatment takes quite some time and requires high memory consumption. Herein, starting from Coulomb law and charge conservation law, we propose an alternative numerical method, namely, two-dimensional finite element method, to solve this problem. All the calculations are now performed in two-dimensional graphene plane, and the usual three-dimensional approximation is not required. To characterize the excitations of graphene magneto-plasmons, the eigenvalue loss spectrum is introduced. Based on this method, graphene magneto-plasmons in graphene rings of four kinds are investigated. The strongest magneto-optic effect is observed in circular ring, which is consistent with its highest rotational symmetry. In all the rings, the lowest dipolar graphene magneto-plasmon always supports symmetric mode splitting, which can be further modified by the interaction between inner edge and outer edge of ring. As the hole size is very small, the edge current confined to the outer edge dominates, and that confined to the inner edge can be ignored; while increasing the hole size, the interaction between these two edges increases, which results in the reduction of the symmetric mode splitting; when the hole size is larger than a critical value, the symmetric mode splitting will disappear.
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Keywords:
- graphene /
- magneto-plasmon /
- finite element method /
- eigenvalue loss spectrum
1. 引 言
等离激元源于自由电子的集体振荡, 是固体中的一种元激发, 具有很多独特的光学性质, 如亚波长约束、局域场增强与突破衍射极限[1-3], 可极大地提升光与物质的相互作用; 作为兼具电子与光子优势的新型载体, 其在物理、化学、材料和生物等领域得到了高度的关注, 广泛应用于光电探测[4-6]、生物传感[7-9]、高分辨成像[10-12]、催化反应[13-15]等. 与其他准粒子(如声子)类似, 振动频率与弛豫时间是等离激元的两个重要指标, 当下的研究多集中于贵金属体系(如Au和Ag), 前者可通过几何结构、尺寸来调节, 以满足不同工作频率的需求, 而后者则严重受制于固有的欧姆损耗[16,17], 相干时间与传输距离较短, 成为其在器件应用方面的主要障碍. 要从根本上解决这一问题, 必须挖掘更好的等离激元材料[18,19], 以替代传统贵金属. 人们尝试了Al[20], Na[21], Cu[22], 以及掺杂的半导体[23]等, 情况有所改善, 但更多有价值的探索仍有待进一步开展.
2004年, 二维(two-dimensional, 2D)材料石墨烯首次由Novoselov等[24]通过机械剥离方法成功制备, 展现出了优异的力学[25-27]、热学[28-30]、光学[31-34]及电学[35-38]特性, 迅即成为理论与应用研究的焦点, 更是掀起了2D材料研究的热潮[39-41]. 作为一种零带隙的半金属, 中性石墨烯不支持等离激元共振, 但当其费米面偏离狄拉克(Dirac)点时[36,37], 便提供了该准粒子激发所需的内在条件[42-45], 且较贵金属有一系列独特的优势: 1) 电磁场高度局域于单原子层的厚度处[46-48]; 2) 共振频率可通过外置电压动态调控[49,50]; 3) 欧姆损耗较小(缘于极高的载流子迁移率) [51-53]; 4) 工作在稀缺的太赫兹及红外波段[54-57]. 相关的工作众多, 汇聚成石墨烯等离激元学(graphene plasmonics)这一新兴的学科方向, 也极大地推动了2D材料光学的发展[58-60]. 毫无疑问, 这里外置电压(即电场)赋予的动态可调性打破了传统等离激元仅可依靠结构调控的桎梏, 是石墨烯等离激元最引人注目的亮点, 为多频段、多功能、可切换光电器件的设计提供了潜在的解决方案. 然而, 该体系中是否还蕴藏着其他调控自由度?
值得注意的是, 与传统的二维电子气(two-dimensional electron gases, 2DEGs)相比, 线性能带结构的石墨烯具有更强的磁场响应, 很多极限条件下的物理效应在常规条件下便可以观测, 如量子霍尔效应[61-63]、分数量子霍尔效应[64-66]和量子自旋霍尔效应[67,68]等; 同时, 体系的光与物质相互作用也受磁场强烈地调制, 即磁光效应(magneto-optic effect, MOE), 比如显著增强的法拉第旋转[69-72]. 鉴于此, 电子的集体振荡以单粒子回旋运动为基础, 其准粒子俗称磁等离激元[73-75], 是MOE与等离激元优势的结合体[76], 而贵金属不支持MOE, 因此和电场一样, 磁场也是石墨烯等离激元独特的调控自由度. 有趣的是, 这两个自由度的调控机理差异很大: 电场改变的是电子的浓度, 而其单粒子的性质不变; 磁场改变的是单粒子的运动状态, 而电子的浓度不变; 两种方式并行不悖, 可灵活搭配以实现最优的等离激元调控[77].
当前, 石墨烯磁等离激元(graphene magneto-plasmon, GMP)的研究多开展于较为简单的石墨烯体系, 如无限大[78-82]、半无限大[83-85]和圆盘结构[86,87]等. 此外, 磁场的存在打破了时间反演对称性, GMP很自然地与非互易[88,89]、拓扑[90-92]等物理现象密切相关. 考虑到不可避免且尤为重要的界面效应, 我们曾探讨了石墨烯半无限薄片[93]、圆盘[94]、正多边形盘[95]、圆环[96,97]中GMP的模式劈裂与杂化等基本物性, 但对界面效应仍缺乏较为全面的认识, 更多深入的探究诸如内外界面的耦合、GMP高阶态的激发等仍有待开展.
鉴于GMP研究大多采用三维(three-dimensional, 3D)近似, 即将2D石墨烯等效成厚度很薄的3D块材, 仿真计算需消耗大量资源. 本文提出一种可行的替代方案, 不再需要3D近似, 所有的计算只限于2D石墨烯面内, 极大地节省了资源、提升了计算效率. 利用该2D计算方法, 讨论了4类石墨烯环中GMP的激发, 分析了内外边界的相互作用对GMP模式劈裂的影响.
2. 理论模型与数值计算方法
2.1 石墨烯的磁光响应
媒质的光学响应由其微观的单粒子跃迁(single-particle transition, SPT)决定, 但可通过宏观的电磁参量如介电函数或电导率表征. MOE体系中SPT发生在不同的朗道能级(Landau level)之间[98-102], 传统抛物线形的2DEG中第
n 个朗道能级的能量为En=(n+1/2)ℏωc , 这里的回旋共振频率(cyclotron resonance frequency, CRF)ωc=eB/m∗ ,m∗ 为电子的有效质量; 而石墨烯中第n 个朗道能级的能量可表示为En=±√nℏωc , 这里ωc=√2vF/lB , 费米速度vF=106 m/s, 磁长度lB=√ℏ/(eB) , 即ωc∝√B , 而不是像传统2DEG中的线性关系; 但在n 很大时(对应高掺杂情况), 石墨烯和2DEG中由于朗道能级不同而导致的SPT差异被逐渐抹平, 石墨烯的磁光面电导率也可用经典的德鲁德(Drude)模型表述[94]:\begin{split} \bar {\bar{{\boldsymbol{\sigma}} }}=\;&\left( \begin{array}{cc}{\sigma }_{xx}& {\sigma }_{xy}\\ -{\sigma }_{xy}& {\sigma }_{yy}\end{array} \right)\\=\;&\dfrac{{e}^{2}{E}_{{\rm{F}}}}{{\rm{\pi }}{\hslash }^{2}\left[{\left(\omega +{\rm{i}}{\tau }^{-1}\right)}^{2}-{\omega }_{{\rm{c}}}^{2}\right]}\\ &\times\left( \begin{array}{cc}{\rm{i}}\left(\omega +{\rm{i}}{\tau }^{-1}\right)& {\omega }_{{\rm{c}}}\\ -{\omega }_{c}& {\rm{i}}\left(\omega +{\rm{i}}{\tau }^{-1}\right)\end{array} \right)\text{. } \end{split} (1) 这里
{\omega }_{{\rm{c}}}=eB/{m}^{*}=eB{v}_{{\rm{F}}}^{2}/{E}_{{\rm{F}}} 为等效的CRF, 而不是石墨烯体系严格的CRF,\tau =\mu {E}_{{\rm{F}}}/\left(e{v}_{{\rm{F}}}^{2}\right) 为弛豫时间, 是表征损耗的唯象参数, 系数\mu 是载流子迁移率. 本文中将这些参数设定为常数(除非单独说明), 如磁场B=5 T, 费米能级{E}_{{\rm{F}}}=0.6 eV, 以及载流子迁移率μ = 10000 cm2·V–1·s–1.2.2 石墨烯磁等离激元的数值模型
基于上述磁光面电导率, 体系的GMP可通过麦克斯韦方程组求解, 一般处理的方法首先是将2D石墨烯近似成非常薄的3D块材[94-97], 比如其厚度是面内尺寸的1%甚至更小, 然后在任意电磁仿真软件中进行建模. 但是这样处理往往有一定的代价, 就是人为引入的这个极小厚度会使剖分的3D网格数量急剧增加, 耗费很多的计算资源与时间. 本文在准静态近似下, 以库仑定律和电荷守恒定律为基础, 构建自洽的2D面积分方程, 并在面内剖分2D网格进行求解. 此前, 我们曾用该方法计算无磁场情况下石墨烯及双层石墨烯体系的等离激元[103-105], 而这里GMP的计算不能简单照搬先前的方案, 需要考虑面电导率中非对角项的贡献. 自洽的2D面积分方程为
\phi \left(\boldsymbol{r}\right)={\phi }^{{\rm{e}}{\rm{x}}{\rm{t}}}\left(\boldsymbol{r}\right)+\frac{1}{4{\rm{\pi }}{\varepsilon }_{{\rm{s}}}}\int \frac{\rho \left({\boldsymbol{r}}'\right)}{\left|\boldsymbol{r}-{\boldsymbol{r}}'\right|}{\rm{d}}{\boldsymbol{r}}', \tag{2a} \rho \left(\boldsymbol{r}\right)=\frac{{\rm{i}}{\sigma }_{xx}}{\omega }{\nabla }^{2}\phi \left(\boldsymbol{r}\right). \tag{2b} 这里的积分和微分运算都只在石墨烯面内, 例如
\nabla ={\partial }_{x}\widehat{\boldsymbol{x}}+{\partial }_{y}\widehat{\boldsymbol{y}} ,{\varepsilon }_{{\rm{s}}} 是周围环境的平均介质常数, 如石墨烯两侧介质的介电常数分别为{\varepsilon }_{1} 和{\varepsilon }_{2} , 则{\varepsilon }_{{\rm{s}}}= \left({\varepsilon }_{1}+{\varepsilon }_{2}\right)/2 . (2a)式是库仑定律的积分形式, 这一点很直观, 需要说明的是, 由这个方程可以求得空间任意位置的电势\phi \left(\boldsymbol{r}\right) , 而不限于石墨烯面内, 但电荷\rho \left(\boldsymbol{r}\right) 只分布于石墨烯面内, 因而(2b)式中的电势\phi \left(\boldsymbol{r}\right) 只限于石墨烯面内, 即石墨烯面内的\phi \left(\boldsymbol{r}\right) 和\rho \left(\boldsymbol{r}\right) 可通过这个方程组自洽求解. (2b)式来自电荷守恒定律, 但为什么其只与面电导率的对角项有关. 解开这个疑团, 需先从石墨烯的面电流入手:\begin{split} \boldsymbol{J}=\;&{J}_{x}\widehat{\boldsymbol{x}}+{J}_{y}\widehat{\boldsymbol{y}}=\left({\sigma }_{xx}{E}_{x}+{\sigma }_{xy}{E}_{y}\right)\widehat{\boldsymbol{x}} \\ & +\left(-{\sigma }_{xy}{E}_{x}+{\sigma }_{xx}{E}_{y}\right)\widehat{\boldsymbol{y}}\\ =\;&-{\sigma }_{xx}\nabla \phi -{\sigma }_{xy}\nabla \times \left[\phi \left(\boldsymbol{r}\right)\widehat{\boldsymbol{z}}\right]{\rm{. }} \end{split} (3) 这里已经利用了
{\sigma }_{xx}={\sigma }_{yy} ; 由此, 不难看出-{\sigma }_{xy}\nabla \times \left[\phi \left(\boldsymbol{r}\right)\widehat{\boldsymbol{z}}\right] 是一个矢量函数的旋度, 其散度恰好为零, 即如(2b)式所示,\nabla \cdot \boldsymbol{J} 只由-{\sigma }_{xx}\nabla \phi 贡献.对于任意形状的石墨烯结构, (2a)式和(2b)式需要离散化求解, 这里沿用之前发展的2D有限元方法(2D finite element method, 2DFEM) [104], 以每个格点上的
\phi \left(\boldsymbol{r}\right) 和\rho \left(\boldsymbol{r}\right) 作为基函数. (2a)式可直接给出关于\phi \left(\boldsymbol{r}\right) 和\rho \left(\boldsymbol{r}\right) 之间关系的第一个矩阵方程, 比如\left(\begin{array}{c}{\phi }_{1}\\ \vdots\\ {\phi }_{N}\end{array}\right)=\left(\begin{array}{c}{\phi }_{1}^{{\rm{e}}{\rm{x}}{\rm{t}}}\\ \vdots\\ {\phi }_{N}^{{\rm{e}}{\rm{x}}{\rm{t}}}\end{array}\right)+{{{\boldsymbol{M}}}}\left(\begin{array}{c}{\rho }_{1}\\ \vdots\\ {\rho }_{N}\end{array}\right)\text{, } (4) N 为格点的数目. 问题的关键在于(2b)式的处理, 需将其转化为弱形式, 即在其两边乘以试探函数\phi \left(\boldsymbol{r}\right) 再积分,\displaystyle\int \phi \left(\boldsymbol{r}\right)\rho \left(\boldsymbol{r}\right){\rm{d}}\boldsymbol{r}=\dfrac{{\rm{i}}{\sigma }_{xx}}{\omega }\displaystyle\int \phi \left(\boldsymbol{r}\right){\nabla }^{2}\phi \left(\boldsymbol{r}\right){\rm{d}}\boldsymbol{r} . 利用恒等式
\nabla \cdot \left(f\boldsymbol{g}\right)=\nabla f\cdot \boldsymbol{g}+f\nabla \cdot \boldsymbol{g} , 等式右边可进行如下约化:\begin{split} &\displaystyle\int \nabla \cdot \left[\phi \left(\boldsymbol{r}\right){\sigma }_{xx}\nabla \phi \left(\boldsymbol{r}\right)\right]{\rm{d}}\boldsymbol{r} - {\sigma }_{xx}\displaystyle\int \nabla \phi \left(\boldsymbol{r}\right) \cdot \nabla \phi \left(\boldsymbol{r}\right)\\ =\;& \displaystyle\int {{\boldsymbol{n}}} \cdot \left[\phi \left(\boldsymbol{r}\right){\sigma }_{xx}\nabla \phi \left(\boldsymbol{r}\right)\right]{\rm{d}}\boldsymbol{l} - {\sigma }_{xx}\displaystyle\int \nabla \phi \left(\boldsymbol{r}\right) \cdot \nabla \phi \left(\boldsymbol{r}\right){\rm{d}}\boldsymbol{r}. \end{split} (5) 等式右边第一项是沿着边界的线积分, n为边界上每处的外法线方向. 要完全求解这个方程必须考虑合理的边界条件, 按照惯例可使法向电流为0,
{\boldsymbol{n}}\cdot{\boldsymbol{J}} = 0 , 即对于电势\phi \left(\boldsymbol{r}\right) 的诺伊曼边界条件(Neumann boundary condition). 在无磁场的情况下, 由于\boldsymbol{J}= -{\sigma }_{xx}\nabla \phi \left(\boldsymbol{r}\right) 第一项线积分正好为0; 但有磁场时, 结合(3)式容易得到\begin{split} & \displaystyle \int \phi \left(\boldsymbol{r}\right)\rho \left(\boldsymbol{r}\right){\rm{d}}\boldsymbol{r} \\ =\;&-\dfrac{{\rm{i}}{\sigma }_{xx}}{\omega }\displaystyle \int \nabla \phi \left(\boldsymbol{r}\right)\cdot \nabla \phi \left(\boldsymbol{r}\right){\rm{d}}\boldsymbol{r}\\ &-\dfrac{{\rm{i}}{\sigma }_{xy}}{\omega }\displaystyle \int \phi \left(\boldsymbol{r}\right)\boldsymbol{n}\cdot \nabla \times \left[\phi \left(\boldsymbol{r}\right)\widehat{\boldsymbol{z}}\right]{\rm{d}}\boldsymbol{l}. \end{split} (6) 该方程离散化的最终形式为
\begin{split} &\left(\begin{array}{ccc}{\phi }_{1}& \cdots & {\phi }_{N}\end{array}\right)\left(\begin{array}{c}{\rho }_{1}\\ \vdots\\ {\rho }_{N}\end{array}\right)=\left(\begin{array}{ccc}{\phi }_{1}& \cdots & {\phi }_{N}\end{array}\right)\\ &\times\left[-\frac{{\rm{i}}{\sigma }_{xx}}{\omega }\boldsymbol{P}-\frac{{\rm{i}}{\sigma }_{xy}}{\omega }\boldsymbol{Q}\right]\left(\begin{array}{c}{\phi }_{1}\\ \vdots\\ {\phi }_{N}\end{array}\right) \text{, } \end{split} 即
\left(\begin{array}{c}{\rho }_{1}\\ \vdots\\ {\rho }_{N}\end{array}\right)=\left[-\dfrac{{\rm{i}}{\sigma }_{xx}}{\omega }\boldsymbol{P}-\dfrac{{\rm{i}}{\sigma }_{xy}}{\omega }\boldsymbol{Q}\right]\left(\begin{array}{c}{\phi }_{1}\\ \vdots\\ {\phi }_{N}\end{array}\right). (7) (4)式和(7)式构成格点上电势
\phi \left(\boldsymbol{r}\right) 与电荷\rho \left(\boldsymbol{r}\right) 的一组自洽方程, 将(7)式代入(4)式可得\left[1+\dfrac{{\rm{i}}{\sigma }_{xx}}{\omega }\boldsymbol{M}\boldsymbol{P}+\dfrac{{\rm{i}}{\sigma }_{xy}}{\omega }\boldsymbol{M}\boldsymbol{Q}\right]\left(\begin{array}{c}{\phi }_{1}\\ \vdots\\ {\phi }_{N}\end{array}\right)=\left(\begin{array}{c}{\phi }_{1}^{{\rm{e}}{\rm{x}}{\rm{t}}}\\ \vdots\\ {\phi }_{N}^{{\rm{e}}{\rm{x}}{\rm{t}}}\end{array}\right). (8) 很显然, 方程左边括号里的部分对应于体系等效的介电函数(dielectric function), 找出其为零的解, 即给出体系自持(self-sustaining)的本征GMP模式, 但考虑到耗散及其矩阵形式, 该方法并不具备可操作性; 这里将使用其本征值损失谱方法, 具体如下: 对任一频率
\omega , 对角化介电函数矩阵, 得到N 个本征值\left({\varepsilon }_{1}, \cdots {, \varepsilon }_{N}\right) 及其本征态, 挑出其中实部趋于零且虚部最小的那个本征值(通过损失函数-{\rm{I}}{\rm{m}}\left({\varepsilon }_{n}^{-1}\right) 的极值), 记为该频率处的谱值, 而在频谱上谱值的峰即对应自持的GMP模式, 如果峰值处对应多个相同的本征值, 则表明该模式为多重简并. 在实施过程中, 矩阵\boldsymbol{M} ,\boldsymbol{P} 以及\boldsymbol{Q} 只由石墨烯结构决定, 因此只需求解一次, 而频率\omega 的依赖则完全包含在系数{\rm{i}}{\sigma }_{xx}/\omega 与{\rm{i}}{\sigma }_{xy}/\omega 中. 除了本征值损失谱, 该2DFEM还方便计算电磁波散射谱、吸收谱等, 对于入射场\left({\phi }_{1}^{{\rm{e}}{\rm{x}}{\rm{t}}}, \cdots {, \phi }_{N}^{{\rm{e}}{\rm{x}}{\rm{t}}}\right) , 求解(8)式可得电势分布\left({\phi }_{1}, \cdots {, \phi }_{N}\right) . 结合(7)式, 求得激发电荷分布\left({\rho }_{1}, \cdots {, \rho }_{N}\right) , 接着计算偶极矩其虚部则对应电磁波吸收; 这里的入射场即每个格点上电势的初始值可随意设置, 如平面波、点源、电子束等, 以满足个性化的计算需求.3. 结果与讨论
通过上述计算方法, 探讨石墨烯圆环、方环、方孔圆环、圆孔方环中GMP的激发. 不失一般性, 首先考虑环的直径或边长
{D}_{{\rm{R}}}=100 nm, 孔的直径或边长{D}_{{\rm{H}}}=50 nm, 其结构如图1所示. 借助开源软件Gmsh[106], 采用三角形微元对结构进行离散化, 在此微元基础上开展微分、积分运算的细节, 请参阅此前的工作[104]. 在实际的计算中, 为了保证结果的收敛性和精度, 往往需要较为精细的网格. 值得注意的是, 最佳的方案并非是直接设置很小的网格(即很小的三角形微元), 而是设置一般尺寸的三角形微元, 然后对每一个三角形微元进行拆分细化(refine by splitting), 即将一个三角形微元拆分成多个三角形微元. 这样一来, 每个三角形的顶点(对应(8)式中的格点)会被更多的微元共用, 即通过引入更少的格点产生更多的微元, 前者决定(8)式中介电函数矩阵的维度, 后者决定计算的精度, 如图1(a)所示的圆环, 5312个格点构成10240个微元, 此时计算结果已完全收敛. 该方案可极大地提高计算效率, 广泛应用于有限元计算.
图 1 石墨烯圆环(a)、方环(b)、方孔圆环(c)、圆孔方环(d)网格划分示意图, 环的直径或边长均为100 nm, 孔的直径或边长均为50 nmFig. 1. Schematic diagrams of mesh generation for graphene circular ring (a), square ring (b), circular ring with square hole (c), and square ring with circular hole (d). The diameteror side length of ring is 100 nm, and that of hole is 50 nm.为了检验算法的可靠性, 首先对比2DFEM与常见3D有限元(3DFEM)的计算结果. 简单起见, 考查石墨烯圆盘中GMP激发, 即图1(a)中孔直径为零的情形. 至于3DFEM, 借助全波仿真软件COMSOL来开展. 构建球形计算区域, 通过完美匹配层(perfectly matched layer, PML)截断, 石墨烯圆盘的厚度设置为0.5 nm, 置于区域中心, 线极化电磁波正入射, 消光截面
{\sigma }_{{\rm{e}}{\rm{x}}{\rm{t}}} 用于表征体系光学响应(详情参阅此前工作[95-97]). 这里,{\sigma }_{{\rm{e}}{\rm{x}}{\rm{t}}} 可借助光学定理求解, 将其除以圆盘的面积S , 即{\sigma }_{{\rm{e}}{\rm{x}}{\rm{t}}}/S , 就得到了消光系数. 该3DFEM在GMP研究中最为常用, 但只适用于探究含有偶极矩的低阶模式, 比如偶极GMP, 其他模式的{\sigma }_{{\rm{e}}{\rm{x}}{\rm{t}}} 均接近于0, 俗称暗模式. 鉴于此, 不妨比较偶极GMP的计算结果(图2). 相比于2DFEM的结果, 3DFEM的谱线出现了细微的红移, 但在磁场诱发的对称模式劈裂(symmetric mode splitting, SMS)方面表现一致(如劈裂的大小{\omega }_{+}-{\omega }_{-} 相同). 图2(b)中劈裂之后的峰高(红色)是蓝色峰高的一半, 源于线极化本征态可分解为等量左/右旋圆极化本征态的线性叠加. 图2(b)中虚线与蓝色曲线相交部分, 给出了磁场B=0 时模式的半峰宽, 对应模式的损耗, 其由辐射损耗和材料吸收损耗构成. 虽然2DFEM由于采用准静态近似而忽略了辐射损耗, 但两种方法给出的半峰宽几乎一样(包括蓝色曲线与红色曲线的峰), 表明材料吸收损耗主导, 这是等离激元共振体系的特性, 在深亚波长GMP体系则尤为明显.
图 2 石墨烯圆盘的偶极GMP谱线, 数据分别来自2DFEM (a)和3DFEM (b), 其中虚线标记了蓝色曲线半峰的高度Fig. 2. Dipolar GMP spectra in graphene disk, obtained from 2DFEM (a) and 3DFEM (b), respectively. The dashed lines mark the position of half maximum of blue curves.与3DFEM对比, 2DFEM框架下的本征值损失谱方法不依赖于模式的偶极矩, 可方便探究各阶GMP的行为. 如图3(a)所示, 角动量
l\ne 0 的模式都会产生SMS, 即每组GMP({\omega }_{-} 和{\omega }_{+} , 红色曲线的峰值)对称分布于磁场为0时等离激元({\omega }_{{\rm{p}}} , 蓝色曲线的峰值)的两侧, 如l=1 的偶极模式、l=2 的四极模式等;l=0 的模式, 俗称轴对称呼吸模式[103], 没有发生SMS, 其共振频率与磁场为0时等离激元的频率基本一致. 这个结果可借助一个简单的图像来理解. 磁场作用于载流子的洛伦兹力宏观表现为沿边界传输的环形电流, 其环向与磁场构成右手定则, 不妨假定为顺时针方向. 对于l\ne 0 的模式, 其共振电流也沿环向, 且两个简并态\pm l 对应的环向恰好相反, 因此顺时针方向模式因与边界环流同向而缩短运动周期, 导致频率增大, 但逆时针方向模式因与边界环流反向而延长运动周期, 导致频率减小, 这里的增减量均等于边界环流即SMS. 对于l=0 的模式, 其共振电流沿径向, 边界环流不会对其产生加速或减速效果, 且该情形下{\omega }_{{\rm{p}}}\gg {\omega }_{{\rm{c}}} , 边界环流对于快速的径向电流可做准静态近似, 即忽略其对后者的影响. 该图像不局限于圆形, 适用于任意形状体系中GMP激发的描述, 详情参阅此前的工作[95].
图 3 石墨烯圆盘(a)和方块(b)的本征值损失谱, 圆盘和方块的直径与边长均为100 nmFig. 3. Eigenvalue loss spectra of graphene disk (a) and graphene square (b). The diameter of disk and the side length of square are both 100 nm.如图3(b)所示, 对于石墨烯方块, 也有5组GMP模式出现在相似的频率范围, 但相比圆盘, 整体发生红移, 且仅有2组伴随着SMS, 被第1条和第5条虚线标注. 从上述分析中可知, 这两个模式类似于圆盘中偶极和四极模式, 而其他3个模式则由径向振动主导. GMP模式上的差异将会影响它们之间的耦合, 圆形或方形作为内边界或外边界时作用也将不同, 因而4类石墨烯环中GMP的值需要进一步深入探究.
在给定的频率范围内, 数值计算的结果如图4所示, 不难发现4类石墨烯环支持的GMP激发有很大不同, 归纳为4点. 1) 圆环拥有最多的4组GMP经历SMS, 体现最强的磁场效应, 这与其体系最高的对称性一致; 2) 方孔圆环仅有2组GMP发生SMS, 因其面积最小, 更多的模式被推至高频; 3) 最低阶偶极GMP均支持SMS, 这由体系的
{C}_{4} 旋转对称性决定; 4) SMS的大小呈不均匀分布, 而非越低阶GMP的SMS越大, 这由环的内外边界对SMS不同的贡献所致, 在均匀的磁场中, 内外边界处的环流恰好反向, 即内边界的出现会抑制SMS, 而抑制的程度取决于GMP在内外边界处的局域场分布.
图 4 4类石墨烯环的本征值损失谱, 椭圆标记了发生SMS的GMP模式Fig. 4. Eigenvalue loss spectra of graphene rings of 4 kinds. The ellipses mark the symmetric mode splitting of those GMPs.上述讨论中提及, 4类石墨烯环中最低阶的偶极GMP都会发生SMS, 但是鉴于内外边界的反向环流, 其耦合作用将极大地调制GMP激发, 这个过程可通过变化孔的尺寸展示. 如图5所示, 在孔很小时(如10 nm), 4类环都具有最大的SMS, 且圆环与方孔圆环、方环与圆孔方环的
{\omega }_{+} 和{\omega }_{-} 都分别相等. 表明该情形下, 外边界的作用主导, 内边界的作用可忽略; 随着孔的尺寸增加,{\omega }_{+} 和{\omega }_{-} 集体红移, 这由其对称的模式分布(即内外边界的激发电荷同号)决定, 与此同时它们的差值, 即SMS明显变小. 值得一提的是, 内外边界的电荷异号, 即典型的反对称模式, 归为径向振动, 不会产生SMS; 在孔很大时(如大于60 nm), 内外边界的反向环流相互抵消,{\omega }_{+} 和{\omega }_{-} 重合, SMS消失. 图6展示了孔的尺寸为60 nm时, 4类石墨烯环中偶极GMP的电势分布图, 印证了关于SMS消失的讨论. 此外, 鉴于该场分布, 亦可通过(6)式判断SMS效应, 如此时内外边界的电势已非常接近, 鉴于其相反的法线方向即对Q矩阵的贡献恰好相反, 导致最终的Q矩阵趋于0.
图 5 4类石墨烯环中偶极GMP模式频率随孔尺寸的演化, 圆圈和菱形分别代表{\omega }_{+} {\omega }_{-} Fig. 5. Resonance frequencies of dipolar GMP as a function of the size of inner hole. The circles and rhombuses are the results of{\omega }_{+} {\omega }_{-} 
图 6 4类石墨烯环中偶极GMP的电势分布图, 孔的尺寸为60 nmFig. 6. Potential distribution of the dipolar GMP in graphene rings of four kinds. The diameter or side length of each ring is 60 nm.4. 结 论
石墨烯微纳结构中的GMP是决定体系磁光性质的重要电磁元激发, 对其开展研究大多仍沿用传统的3D建模方案, 即将2D石墨烯近似成厚度很小的3D块材, 代价是要消耗极大的计算资源. 本文基于库仑定律和电荷连续方程提出用于计算石墨烯体系中GMP的2DFEM, 通过本征值损失谱表征GMP的模式分布. 借助2DFEM, 探讨了4类石墨烯环中GMP 物性, 圆环表现出最强的磁光响应, 与其最高的对称性一致; 最低阶的偶极GMP都支持SMS, 当孔的尺寸很小时, 其对SMS的影响可以忽略, 随着孔的尺寸变大, 内外边界的相互作用增强并最终导致SMS消失. 该2DFEM具有一定的普适性, 可用于任意形状石墨烯结构中GMP的计算, 亦可拓展到其他2D材料, 为GMP的深入研究提供高效便捷的方案.
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图 1 石墨烯圆环(a)、方环(b)、方孔圆环(c)、圆孔方环(d)网格划分示意图, 环的直径或边长均为100 nm, 孔的直径或边长均为50 nm
Figure 1. Schematic diagrams of mesh generation for graphene circular ring (a), square ring (b), circular ring with square hole (c), and square ring with circular hole (d). The diameteror side length of ring is 100 nm, and that of hole is 50 nm.
图 2 石墨烯圆盘的偶极GMP谱线, 数据分别来自2DFEM (a)和3DFEM (b), 其中虚线标记了蓝色曲线半峰的高度
Figure 2. Dipolar GMP spectra in graphene disk, obtained from 2DFEM (a) and 3DFEM (b), respectively. The dashed lines mark the position of half maximum of blue curves.
图 3 石墨烯圆盘(a)和方块(b)的本征值损失谱, 圆盘和方块的直径与边长均为100 nm
Figure 3. Eigenvalue loss spectra of graphene disk (a) and graphene square (b). The diameter of disk and the side length of square are both 100 nm.
图 4 4类石墨烯环的本征值损失谱, 椭圆标记了发生SMS的GMP模式
Figure 4. Eigenvalue loss spectra of graphene rings of 4 kinds. The ellipses mark the symmetric mode splitting of those GMPs.
图 5 4类石墨烯环中偶极GMP模式频率随孔尺寸的演化, 圆圈和菱形分别代表
{\omega }_{+} 与{\omega }_{-} 模式Figure 5. Resonance frequencies of dipolar GMP as a function of the size of inner hole. The circles and rhombuses are the results of
{\omega }_{+} and{\omega }_{-} , respectively.
图 6 4类石墨烯环中偶极GMP的电势分布图, 孔的尺寸为60 nm
Figure 6. Potential distribution of the dipolar GMP in graphene rings of four kinds. The diameter or side length of each ring is 60 nm.
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