应变石墨烯表面电子动力学的数值准共形变换
Numerical quasi-conformal transformations for electron dynamics on strained graphene surfaces
摘要：
利用弯曲时空中的狄拉克方程对一般静态应变石墨烯表面中低能电子的动力学行为进行了数学建模。在笛卡尔坐标系中，可以直接获得表面的参数化，但对于一般的表面变形，所得到的狄拉克方程是复杂的。引入两种不同的策略来简化这个问题：对角度量近似和变量的等温坐标的变化。这些坐标是从Beltrami方程表征的准保角变换获得的，Beltrami方程的解给出了两个坐标系之间的映射。为了实现这第二个战略，最小二乘有限元数值方案被引入到解决贝尔特拉米方程。狄拉克方程，然后通过一个精确的伪谱数值方法在伪厄米特表示，被赋予明确的酉演化和保守的范数。这两种方法进行了比较，并施加到高斯形状的石墨烯表面变形的电子散射。结果表明，电子波包可以被这些局域应变区聚焦。
英文摘要：
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 quasi-conformal transformations characterized by the Beltrami equation, whose solution gives the mapping between both coordinate systems. To implement this second strategy, a least square finite-element numerical scheme is introduced to solve the Beltrami equation. The Dirac equation is then solved via an accurate pseudo-spectral 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.
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