The dynamics of low-energy electrons in general static strained graphene surface is modelled mathematically by the Dirac equation in curved space-time. In Cartesian coordinates, a parametrization of the surface can be straightforwardly obtained, but the resulting Dirac equation is intricate for general surface deformations. Two different strategies are introduced to simplify this problem: the diagonal metric approximation and the change of variables to isothermal coordinates. These coordinates are obtained from quasiconformal transformations characterized by the Beltrami equation, whose solution gives the mapping between both coordinate systems. To implement this second strategy, a least-squares finite-element numerical scheme is introduced to solve the Beltrami equation. The Dirac equation is then solved via an accurate pseudospectral numerical method in the pseudo-Hermitian representation that is endowed with explicit unitary evolution and conservation of the norm. The two approaches are compared and applied to the scattering of electrons on Gaussian shaped graphene surface deformations. It is demonstrated that electron wave packets can be focused by these local strained regions.

中文翻译：

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应变石墨烯表面上电子动力学的数值拟保形变换

一般静态应变石墨烯表面中的低能电子动力学是通过Dirac方程在弯曲时空中进行数学建模的。在笛卡尔坐标系中，可以直接获得表面的参数化，但是所得的Dirac方程对于一般的表面变形来说是复杂的。引入了两种不同的策略来简化此问题：对角线度量逼近和将变量更改为等温坐标。这些坐标是从以Beltrami方程为特征的拟形变换中获得的，其解给出了两个坐标系之间的映射。为了实现第二种策略，引入了最小二乘有限元数值方案来求解Beltrami方程。然后，通过伪赫米特表示中的精确伪谱数值方法，求解Dirac方程，并赋予其显式evolution演化和范数守恒。比较了这两种方法，并将其应用于高斯形石墨烯表面形变上电子的散射。已经证明，电子波包可以被这些局部应变区域聚焦。