无序介质中自由费米子经典电导率理论在原子尺度上的准确性
Accuracy of Classical Conductivity Theory at Atomic Scales for Free Fermions in Disordered Media
摘要：
对更小的电子元件的日益增长的需求最近引发了对在原子尺度附近经典电导率理论崩溃的兴趣，在原子尺度下量子效应应该占主导地位。在2012年，实验测量的电阻的纳米线在硅掺杂磷原子表明，量子效应的电荷传输几乎消失的纳米线的长度大于几个纳米，即使在非常低的温度（4.2 K）。我们在数学上证明，对于非相互作用的晶格费米子与无序，其（经典）宏观值周围的微观电流密度的量子不确定性被抑制，指数快速相对于外部电场施加的晶格区域的体积。这与上述实验观察结果一致。无序是由随机的外部电位以及随机的，复值的，跳跃幅度。著名的紧束缚安德森模型是这里考虑的一般情况的一个特殊例子。我们的数学分析是基于Combes-Thomas估计，Akcoglu-Krengel遍历定理和大偏差形式主义，特别是Gärtner-Ellis定理。
英文摘要：
The growing need for smaller electronic components has recently sparked the interest in the breakdown of the classical conductivity theory near the atomic scale, at which quantum effects should dominate. In 2012, experimental measurements of electric resistance of nanowires in Si doped with phosphorus atoms demonstrate that quantum effects on charge transport almost disappear for nanowires of lengths larger than a few nanometers, even at very low temperature (4.2K). We mathematically prove, for non-interacting lattice fermions with disorder, that quantum uncertainty of microscopic electric current density around their (classical) macroscopic values is suppressed, exponentially fast with respect to the volume of the region of the lattice where an external electric field is applied. This is in accordance with the above experimental observation. Disorder is modeled by a random external potential along with random, complex-valued, hopping amplitudes. The celebrated tight-binding Anderson model is one particular example of the general case considered here. Our mathematical analysis is based on Combes-Thomas estimates, the Akcoglu-Krengel ergodic theorem, and the large deviation formalism, in particular the Gärtner-Ellis theorem.
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