單層石墨烯(single layered graphene)是原子厚度等級的材料，其晶格結構呈蜂巢狀，由碳原子以sp2混成軌域緊密排列呈平面六角形，屬於零能隙半導體或稱類金屬(semi-metal)，不同於矽半導體需要三、五族元素摻雜才能改變電性，單層石墨烯被外加電場或磁場，就能調變電性，因此有研究團隊製作石墨烯電子元件，除了石墨烯DC導電度可被外加電場調控，光源頻率變化亦會造成石墨烯AC導電度改變。有研究團隊用紅外線光源(infrared microscopy)得到石墨烯的實虛部導電度隨著光源頻率變化。有研究團隊利用TM模態表面電漿波只能傳遞在石墨烯虛部導電度是負值區域，模擬出石墨烯奈米帶的效果，引起我們注意石墨烯表面電漿，於是本文嘗試模擬單層石墨烯波導(single layered graphene waveguide)。
我們參考文獻方法去描述單層石墨烯的二維導電度，然後考慮其電流項於馬克士威-安培定律，橫磁模態(Transverse Magnetic Mode)波傳遞在未加電壓的單層石墨烯有97.74%穿透、2.246%吸收與0.014%反射；而加電壓的單層石墨烯其吸收係數與波長有關，當外加電壓為0.6eV 時在1.2m有最大吸收值6.633%。我們給定初始傳播常數，利用Compact FDTD計算單層石墨烯波導管的時域電場，藉由時域電場的富立葉轉換頻譜去找出對應頻率，因此獲得色散曲線。我們得知石墨烯intraband導電度對應的介電係數有金屬之Drude模型，所以模擬石墨烯intraband波導管，結果顯示厚度愈薄的石墨烯intraband波導管，對稱與反對稱模態的色散曲線愈分開，對稱模態色散曲線愈靠近傳播常數軸，也就是說，等效波長縮短，此特性有潛力應用於高解析度光學顯微技術。

Single layered graphene is an atomically thick material with sp2 hybridized carbon atoms tightly packed into a 2D honeycomb lattice, and behaves as a zero-gap semiconductor or semi-metal. The electronic properties of single layered graphene can be controlled by externally applied voltage or magnetic field. This tunability distinguishes graphene from silicon semiconductors and noble metals. Infrared microscopy measurement shows that the real part and imaginary part of the AC conductivity in the single layered graphene exhibits plasmonics behavior near the infrared regime. The transverse magnetic (TM) mode surface plasmonic wave is allowed to propagate in graphene nanoribbons. Those recent interesting findings motivate us to simulate the single layered graphene waveguide.
2D conductivity of a single layered graphene can be obtained from Kubo formula and applied to Maxwell-Ampere’s law with auxiliary differential equation method in finite-difference time-domain(FDTD) method. When no electrical gating is applied, single layered graphene shows 2.246% absorption. However, when electrical gating voltage is applied, the absorption coefficient of a single layered graphene varies with frequency. Using Compact FDTD, the dispersion relations of single layered graphene waveguide were obtained. When the thickness of the graphene plasmonic waveguide varies, the thinner the single layered graphene waveguide, the further splitting occurs for the waveguide dispersion curve between the symmetric and the anti-symmetric modes. The dispersion curve for symmetric mode is closer to the propagation axis, in other words, the effective wavelength is smaller. This property has potential applications for high resolution optical microscopy technology.

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