石墨烯E-2g和A-1g振动模的应力解释：理论分析
Stress intrepretation of graphene E-2g and A-1g vibrational modes: theoretical analysis
摘要：
在这里，我们只关注一个石墨烯环，并研究石墨烯的E2 g和A1 g振动模式对应于哪个应力张量分量。这些模式通常分别与G峰和D峰相关，并且与沿试样的应力分布密切相关。我们采用Admal和Tadmor（[1]）的理论框架来定义Cauchy应力张量，并在这个框架中引入E2 g和A1 g作为适当的扰动。我们使用这些扰动的应力张量表达式和评估的应力张量分量与每个振动模式。这种方法虽然本质上是定性的，但它结合了所有主要的物理学，并揭示了当石墨烯静止时（即没有外部施加载荷），E2 g和A1 g振动模式应该与剪切和轴向应力分量有关。为了使我们的框架更接近于更具体的结果，我们评估了对应于静止时的E2 g和A1 g模式的对势的瞬时Hardy应力张量。我们的分析扩展到考虑到一个应用的外部拉伸场。假设扶手椅方向沿着x轴，当沿着扶手椅方向施加张力时，轴向$σ$ 11的应力分量比$σ$ 12、$σ$ 22占优势，后者具有较小的阶数。当张力沿着Z字形方向时，轴向$σ$ 22应力分量支配$σ$ 12、$σ$ 11。当张力处于扶手椅和Z字形方向之间的任意方向时，所有应力分量都具有相同的量级，即使在小应变下也应该全部考虑。
英文摘要：
We here focus on only one graphene ring and examine to which stress tensor components the E2g and the A1g vibration mode of graphene correspond. These modes are typically related with the G-peak and the D-peak, respectively, and are strongly related to the stress distribution along the specimen. We adopt the theoretical framework of Admal and Tadmor ([1]) for the macroscopic definition of the Cauchy stress tensor and we introduce into this framework the E2g and the A1g as appropriate perturbations. We use these perturbations to the stress tensor expression and evaluate which stress tensor components are related to each vibrational mode. This approach, though qualitative in nature, incorporates all the main physics and reveals that E2g and A1g vibration modes should be related to shear as well as axial stress components when graphene is at rest (i.e. no external applied loading). To bring our framework closer to more concrete results, we evaluate the instantaneous Hardy stress tensor for a pair potential which correspond to the E2g and A1g modes at rest. Our analysis expands to take into account an applied external tensile field. Taking the armchair direction to be along the x-axis, when tension applies along the armchair direction, it is the axial $σ$11 stress component which dominates over $σ$12, $σ$22, which are of smaller order. When tension is along the zig-zag direction, it is the axial $σ$22 stress component that dominates over $σ$12, $σ$11. When tension is at an arbitrary direction between the armchair and the zig-zag direction, all stress components are of the same order and should all be taken into account even at small strains.
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