二维有限元方法研究石墨烯环中磁等离激元

王伟华

doi:10.7498/aps.72.20222467

摘要

石墨烯等离激元是决定石墨烯光学性质的重要元激发, 拥有一系列优异的特性, 其通过外置电场的动态可调性最引人注目; 石墨烯具有很强的磁场响应(如室温观测的量子霍尔效应), 因而磁场可作为一个新的调控自由度, 形成的准粒子叫作石墨烯磁等离激元. 鉴于石墨烯的二维属性, 石墨烯磁等离激元的研究大多采用三维近似, 即将石墨烯等效成厚度很薄的三维块材, 该处理方案需消耗大量的计算资源. 本文在准静态近似下, 围绕库仑定律和电荷守恒定律, 构建了高效的二维有限元方法, 自洽地求解石墨烯面内的积分微分方程, 并提出本征值损失谱表征准粒子的激发. 利用二维有限元方法, 探讨了4类石墨烯环中磁等离激元的激发; 最低阶的偶极共振都支持磁等离激元的对称劈裂, 在孔很小时, 其对模式劈裂的影响可忽略, 但当孔的尺寸变大时, 内外边界的相互作用将抑制模式劈裂, 并最终导致其消失.

关键词:

Abstract

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.

Keywords:

作者及机构信息

王伟华

中国矿业大学材料与物理学院, 徐州　221116

通信作者: 王伟华, wh.wang@outlook.com

基金项目: 国家自然科学基金(批准号: 12174440)资助的课题.

Authors and contacts

Wang Wei-Hua

School of Material Science and Physics, China University of Mining and Technology, Xuzhou 221116, China

Corresponding author: Wang Wei-Hua, wh.wang@outlook.com

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 12174440).

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施引文献

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

Fig. 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.

下载: 全尺寸图片幻灯片

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.

下载: 全尺寸图片幻灯片

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

Fig. 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.

下载: 全尺寸图片幻灯片

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.

下载: 全尺寸图片幻灯片

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

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 }_{+} $ and $ {\omega }_{-} $, respectively.

下载: 全尺寸图片幻灯片

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

Fig. 6. Potential distribution of the dipolar GMP in graphene rings of four kinds. The diameter or side length of each ring is 60 nm.

下载: 全尺寸图片幻灯片

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