Dick Rainer
Physics & Engineering Physics, University of Saskatchewan, 116 Science Place, Saskatoon, SK S7N 5E2 Canada.
Nanoscale Res Lett. 2010 Oct;5(10):1546-54. doi: 10.1007/s11671-010-9675-1. Epub 2010 Jul 2.
We consider electrons in the presence of interfaces with different effective electron mass, and electromagnetic fields in the presence of a high-permittivity interface in bulk material. The equations of motion for these dimensionally hybrid systems yield analytic expressions for Green's functions and electromagnetic potentials that interpolate between the two-dimensional logarithmic potential at short distance, and the three-dimensional r(-1) potential at large distance. This also yields results for electron densities of states which interpolate between the well-known two-dimensional and three-dimensional formulas. The transition length scales for interfaces of thickness L are found to be of order Lm/2m() for an interface in which electrons move with effective mass m(), and Lϵ()/2ϵ for a dielectric thin film with permittivity ϵ() in a bulk of permittivity ϵ. We can easily test the merits of the formalism by comparing the calculated electromagnetic potential with the infinite series solutions from image charges. This confirms that the dimensionally hybrid models are excellent approximations for distances r ≳ L/2.
我们考虑存在具有不同有效电子质量界面时的电子，以及在块状材料中存在高介电常数界面时的电磁场。这些维度混合系统的运动方程给出了格林函数和电磁势的解析表达式，它们在短距离的二维对数势与长距离的三维r⁻¹势之间进行插值。这也得出了态密度的结果，该结果在著名的二维和三维公式之间进行插值。对于厚度为L的界面，当电子以有效质量m移动时，过渡长度尺度约为Lm/2m；对于在介电常数为ϵ的块状材料中具有介电常数ϵ的介电薄膜，过渡长度尺度约为Lϵ/2ϵ。我们可以通过将计算出的电磁势与镜像电荷的无穷级数解进行比较，轻松检验该形式体系的优点。这证实了对于距离r≳L/2，维度混合模型是很好的近似。
