扭角石墨烯的超導機制一直是凝態物理中的未解之謎。在本論文中，我的目標是找出扭角石墨烯中可能存在的非傳統超導相。我們特別關注由隨機相位近似修正的排斥交互作用所引起的超導。藉由平均場論，我們證明手徵p+ip和手徵d+id 的超導配對都可以存在。更甚者，我們也分析這些超導配對如何隨溫度及由化學能決定的電子數量和彼此競爭。我們可以證明平均場的相圖起源於p波,次晶格之間的單重態,同層配對 和p/d 波,同次晶格,同層配對的競爭關係。我們的平均場論為系統處在超導相時的庫柏對波函數提供完整的描述。

在了解所有可能的配對後，我們需要計算由Kosterlitz-Thouless 相變決定的臨界溫度。藉由線性響應理論且只考慮近平坦能帶，我們在化學位能-配對強度的參數空間中計算傳統項幾何項的貢獻。我們區分出在化學位能-配對強度的參數空間裡被傳統項或幾何項主導的區域。這些結果不只為對實驗有助益的超導臨界溫度給出定量估計外，也顯示在不同的參數範圍內，超流密度的物理起源可能會有所差別。

The mechanism of superconductivity observed in twisted bilayer graphene remains an unsolved puzzle in condensed matter physics. In this thesis, we aim to find out possible unconventional superconducting phases in twisted bilayer graphene . Specifically, we focus on superconductivity arising from the repulsive interaction corrected by random phase approximation (RPA). In the mean field theory, we show that the chiral p+ip and chiral d+id superconducting pairing are dominant superconducting instabilities. Furthermore, we analyze how these pairing symmetries compete with each other by changing temperature and number of electrons through the chemical potential. It is shown that the mean-field phase diagram results from the competition between p/d-wave intra-sublattice/intra-layer pairing and p-wave inter-sublattice(singlet)/intra-layer pairing. Our mean field theory provides a compact description of the Cooper pair wavefunction when the system is superconducting.

Knowing all possible mean-field pairing states, in 2D, we need to compute the superfluid density to obtain the superconducting transition temperature determined by the Kosterlitz-Thouless transition. By employing the linear response theory and focusing on flat bands, we evaluate the conventional and geometrical contributions to the superfluid density in the chemical potential - pairing amplitude space. Regions in the chemical potential - pairing amplitude space that are dominated either by geometric term or conventional terms are identified. These results not only provide the estimation for superconducting transition temperatures useful for experiments, but also reveal different physical origin of superconducting transitions in different parameter region.

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