從石墨烯的電導率可先分為Inter-band與Intra-band兩個能帶，分析哪個能帶個別受到低頻或高頻的影響，並了解在何時才會有表面電漿的存在，設定觀察的範圍，在程式模擬的操作上判斷是否可以直接使用FDTD演算法直接帶入，而Intra-band的部分可以使用Drude-mode來做fitting，而分析Inter-band的式子，發現其並不是jω多項式，沒有辦法直接代入，所以只能用本實驗室所開發的一個近似模型來模擬Inter-band的能帶，此模型利用ADE(auxiliary differential equations)來代入FDTD來模擬，透過Dispersion的模擬也能證明此模型與Inter-band的特性極為相符，一開始我們只針對Hard-boundary做模擬，後來我們加上電極，調控石墨烯的nanoribbon，針對特定的ribbon寬度去模擬，我們稱為Soft-boundary，不僅分析上述兩種狀態下的Graphene 其模態和特性，我們也在分析在nanoribbon加上介電質(n=2)，來增加其空間中的折射率並觀察其影響，最後在應用上為了實現量子級聯雷射(Quantum Cascade Laser)，其為在中、遠紅外範圍的長波段雷射，為了有效將長波段雷射壓縮在共振腔中，因此我們設計一個Ring Resonator利用高等效的折射率來達到我們的目的，並觀察分析其環形的共振模態。

In this thesis, we analyze the possibility of using graphene as a ring resonator for Quantum Cascade laser. The conductivity of graphene could be divided into intra-band and inter-band. We develop methods to model graphene conductivity for Finite-Difference Time-Domain method. As for the Intra-band, it can be simulated by Drude-model. However, the inter-band cannot be incorporated directly in FDTD since it’s not a polynomial form of jω. Therefore, we used forth-order Pade approximation method to model graphene interband conductivity in FDTD. To design the ring resonator using graphene, we first use the hard-boundary for etched graphene . After that, we use electric field gated graphene to control the nanoribbon of graphene that’s so-called the soft-boundary for graphene conductivity
We not only analyze the above two situations of Graphene, including its mode and characteristics but also add the dielectric beneath nanoribbon to provide the electric gate. In order to achieve Quantum Cascaded Laser which work in the mid-infrared and far infrared, we designed a Ring Resonator with using high N_eff to confine dozens of micron long wavelength THz mode in the submicron graphene ring resonator effectively.

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