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空间反演对称性和时间反演对称性保护着狄拉克锥,但当任意一个被破坏时,狄拉克简并将被打开,产生拓扑带隙。
量子霍尔效应基于时间反演对称性被破坏,用第一陈数C来描述此系统情况下的拓扑性质,其表征整数量子霍尔效应的拓扑不变量,它只能在能隙闭合的地方才会改变。引入Z2不变量[10]表征具有时间反演对称性系统的拓扑性质:
$ {Z_2} = \frac{1}{{2{\rm{ \mathsf{ π} }}}}\left[ {\oint\limits_{\partial {B^ + }} A (\mathit{\boldsymbol{k}}){\rm{d}}l - \int_{{B^ + }} \mathit{\boldsymbol{\Omega }} (\mathit{\boldsymbol{k}}){\rm{d}}{k^2}} \right]\, \bmod \, 2 $
(1) 式中,k是波矢, A(k)是贝里联络,Ω(k)是贝里曲率,B+表示半个布里渊区[21],∂B+为沿着半个布里渊区B+的边界,dl表示在半个布里渊区里的路径积分微元,mod 2表示除以2取余数,Z2=0为普通绝缘体,Z2=1为拓扑绝缘体。
在霍尔电导实验中,垂直电导Rxx=零,霍尔电阻Rxy确定为定值量子化常数RH=h/ve2,v为填充数。对于整数量子霍尔效应,霍尔电导为:
$ {\sigma _{xy}} = N\frac{{{e^2}}}{h} = N/(25812.807572\Omega ) $
(2) 式中,h是普朗克常数,e是电子电荷,N为正整数。
本文中采取调节石墨烯圆盘的直径大小来破坏晶格的时间反演对称性来打开狄拉克点,获得拓扑带隙,产生具有免疫结构缺陷和背向散射的拓扑边界保护态。
为控制同一周期内两个石墨烯圆盘的化学势一致,运用单层石墨烯上横向磁(transverse magnetic,TM)模的色散关系,在的非延迟体系中β»k0,即:
$ \beta = {\varepsilon _0}\frac{{{\varepsilon _{{\rm{air }}}} + {\varepsilon _{{\rm{Si}}{{\rm{O}}_2}}}}}{2}\frac{{2{\rm{i}}\omega }}{{{\sigma _{\rm{g}}}}} $
(3) 式中,εair=1,εSiO2=3.9,分别是空气和二氧化硅的介电常数,对应于本研究中的表层材料和基底材料,ε0是自由空间的真空介电常数,k0=ω/c为自由空间中的波数,β是在表面等离子体激元在石墨烯层上的传播常数,σg为石墨烯的表面电导率,ω是等离激元的角频率,c是光速。
而石墨烯的带间电导率为:
$ {\sigma _{{\rm{inter }}}} = \frac{{i{e^2}}}{{4{\rm{ \mathsf{ π} }}\hbar }}\ln \left[ {\frac{{2\left| {{\mu _{\rm{c}}}} \right| - \hbar (\omega + {\rm{i}}/\tau )}}{{2\left| {{\mu _{\rm{c}}}} \right| + \hbar (\omega + {\rm{i}}/\tau )}}} \right] $
(4) 式中,$\hbar $是还原的普朗克常数,μc是化学势,τ是电子动量弛豫时间。
由上式可知,可通过施加外部电压来控制两个石墨烯圆盘的化学势保持相同。基于控制变量法,仅调节两个石墨烯圆盘的直径大小来破坏时间反演对称性。
周期性石墨烯盘表面拓扑边界传输态研究
Research on topological boundary transmission state on the surface of periodic graphene disk
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摘要: 为了提高光学器件的传输性能,推动微型光学器件及大规模集成光路的发展,提出了一种新型2-D石墨烯等离子体光子晶体结构。通过周期性排列蜂窝状的石墨烯盘,调整一个周期内两个石墨烯圆盘的直径,打开狄拉克点;基于有限元法,运用COMSOL仿真光场传播、计算光子带隙,打破时间反演对称性,实现能带拓扑效应。结果表明,该设计的晶格常数等均处于纳米量级,相较于自由空间波长减小近30倍,具有小型化、高集成化等优势;该结构可在15.3THz~15.8THz频段范围进行动态调制。此研究为设计具有鲁棒性的纳米量级光子器件提供了参考,有望在波导频率的调控、光开关等领域拓展应用。Abstract: In order to improve the transmission performance of optical devices and promote the development of micro-optical devices and large-scale integrated optical circuits, a new type of 2-D graphene plasma photonic crystal structure was proposed. By periodically arranging the honeycomb-shaped graphene discs, the diameter of two graphene discs in a period was adjusted to open the Dirac point; based on the finite element method, the propagation of the light field was simulated by COMSOL and the photonic band gap was then calculated. Through breaking the time inversion symmetry, the band topology effect was achieved. The results show that the lattice constants of this designed are all on the order of nanometers, which is nearly 30 times smaller than the free space wavelength with the advantages of miniaturization and high integration. The structure can be dynamically modulated in the 15.3THz~15.8THz frequency range. This research provides a reference for the design of robust nano-scale photonic devices, and is expected to be applied to waveguide frequency control, optical switches and other fields.
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Key words:
- optoelectronics /
- graphene /
- diameter /
- photonic crystals /
- band topology effect
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[1] LU L, JOANNOPOULOS J D, SOLJA ČIĆM. Topological photonics[J]. Nature Photonics, 2014, 8(11): 821-829. doi: 10.1038/nphoton.2014.248 [2] NOVOSELOV K S. Electric field effect in atomically thin carbon films[J]. Science, 2004, 306(5696): 666-669. doi: 10.1126/science.1102896 [3] XIAO S Sh. Graphene nanophotonics: From fundamentals to applications[C]// 2016 Progress in Electromagnetic Research Symposium (PIERS). New York, USA: IEEE, 2016: 3130. [4] BO X, ZHOU M, GUO L. Electrochemical sensors and biosensors based on less aggregated graphene[J]. Biosensors & Bioelectronics, 2017, 89(Pt 1): 167-186. [5] CINTI S, ARDUINI F. Graphene-based screen-printed electrochemical (bio)sensors and their applications: Efforts and criticisms[J]. Biosensors & Bioelectronics, 2017, 89(Pt 1): 107-122. [6] MAIER S A, BARCLAY P E, JOHNSON T J, et al. Low-loss fiber accessible plasmon waveguide for planar energy guiding and sensing[J]. Applied Physics Letters, 2004, 84(20): 3990-3992. doi: 10.1063/1.1753060 [7] WANG L. Research on optical properties of precious metal nanostructures and enhancement of optical effects by surface plasmons[D]. Hangzhou: Zhejiang University, 2011: 1-128(in Chinese). [8] REN M X, XU J J. Principle and application of surface plasma stimulus to enhance nonlinearity[J]. Laser & Optoelectronics Progress, 2013, 50(8): 080002(in Chinese). [9] WEI Zh N. Nano-plasma catalytic sensing research[D]. Chongqing: Chongqing University, 2018: 1-139 (in Chinese). [10] OCHIAI T, ONODA M. Photonic analog of graphene model and its extension—Dirac cone, symmetry, and edge states[J]. Physical Review, 2012, B80(15): 155103. [11] KHANIKAEV A B, HOSSEIN MOUSAVI S, TSE W K, et al. Photonic topological insulators[J]. Nature Materials, 2013, 12(3): 233-239. doi: 10.1038/nmat3520 [12] MA T, KHANIKAEV A B, MOUSAVI S H, et al. Guiding electromagnetic waves around sharp corners: Topologically protected photonic transport in metawaveguides[J]. Physical Review Letters, 2015, 114(12): 127401. doi: 10.1103/PhysRevLett.114.127401 [13] CONSTANT T J, HORNETT S M, CHANG D E, et al. All-optical generation of surface plasmons in graphene[J]. Nature Physics, 2015, 12(2): 124-127. [14] SLOBOZHANYUK A P, KHANIKAEV A B, FILONOV D S, et al. Experimental demonstration of topological effects in bianisotropic metamaterials[J]. Scientific Reports, 2016, 6(1): 22270-22275. doi: 10.1038/srep22270 [15] MA R K, ZHANG Y Ch, FANG Y T. Broadband THz absorbers based on graphene and 1-D photonic crystal[J]. Laser Technology, 2017, 41(5): 723-727(in Chinese). [16] LU L, WANG Z. Topological one-way fiber of second Chern number[J]. Nature Communications, 2016, 9(1): 821-829. [17] SONG Z D. Topological photonic crystal design and research based on graphene[D]. Xi'an: University of Chinese Academy of Sciences (Xi'an Institute of Optical Precision Machinery, Chinese Academy of Sciences), 2018: 1-70(in Chinese). [18] GAO Y F, JIANG Zh, LIU K Y, et al. Optical waveguide for photonic crystal topological boundary state photonic spin guidance mechanism: CN110007398A[P]. 2019-07-12(in Chinese). [19] HAN J, GAO Y, JIAO W Y, et al. Mid-infrared plasmon control based on graphene nanoribbons[J]. Chinese Journal of Optics, 2020, 13(3): 627-636(in Chinese). [20] BI L, YANG Y C, QIN J, et al. A silicon-based integrated optical isolation device based on topological protection mechanism: CN110941109A[P]. 2020-03-31(in Chinese). [21] LIANG FU, KANE C L. Time reversal polarization and a Z2 adiabatic spin pump[J]. Physical Review, 2006, B74(19): 195312.