二维耦合无序体系量子扩散的周期性振荡

Periodic Oscillations of Quantum Diffusion in Two-Dimensional Coupled Disordered Systems

摘要: 本文首先介绍了二维材料与耦合机制，随后基于紧束缚近似理论研究二维耦合无序量子扩散现象，运用解析及数值的方法证实了二维耦合体系横向及纵向传播中的同步量子扩散周期性振荡。研究表明：双层无序石墨烯体系周期性振荡对无序度承受能力相对双层无序四方晶格体系更强。边界无序体系周期性振荡任意无序度下均存在。当无序仅处于系统边界，$P_k\left(t\right)$ ，$d_k\left(t\right)$ 振荡在波包到达边界后开始以$y_k\left(t\right)=0.5\pm 0.5e^{-t/t_0}cos\left(2vt+\varphi \right)$ ，$y_k\left(t\right)=P_k\left(t\right)$ 或${\overline{d}}_k^2\left(t\right)$ 函数衰减，衰减速度符合$1/t_0~W^{1.75}$ ，较双层无序系统衰减更慢。边界无序系统周期性衰减速度随$W$ 存在由增加到降低的转变，强无序下出现$W$ 越大衰减速度越慢的反常量子扩散现象。边界无序体系中量子扩散周期性振荡始终存在。研究表明：这一周期性振荡是由层间量子跳跃导致，是各层无差异能谱的结果，与体系形状、大小、边界条件、以及格点能大小无关，且适用于不同原子结构体系。无序系统中层间能谱差异性导致了周期性振荡的衰减，其衰减速度与频率可分别通过体系无序度$W$ 与层间耦合强度$u$ 进行调控。

Abstract: This paper first introduces two-dimensional materials and their coupling mechanisms. It then investigates the phenomenon of two-dimensional coupled disordered quantum diffusion based on the tight-binding approximation theory. Using both analytical and numerical methods, the study confirms the existence of synchronous quantum diffusion periodic oscillations in the transverse and longitudinal propagation of two-dimensional coupled systems. The results indicate that the periodic oscillations in a bilayer disordered graphene system exhibit greater resilience to disorder compared to a bilayer disordered square lattice system. Periodic oscillations persist in boundary disordered systems across various levels of disorder. When disorder is confined solely to the system’s boundary, the oscillations of $P_k\left(t\right)$ and $d_k\left(t\right)$ commence after the wave packet reaches the boundary, described by $y_k\left(t\right)=0.5\pm 0.5e^{-t/t_0}cos\left(2vt+\varphi \right)$ . Here, $y_k\left(t\right)$ can be either $P_k\left(t\right)$ or ${\overline{d}}_k^2\left(t\right)$ and the decay of the function follows the relationship $1/t_0~W^{1.75}$ , indicating a slower decay rate compared to bilayer disordered systems. The decay rate of periodic oscillations in boundary disordered systems transitions from increasing to decreasing as the disorder strength $W$ varies. An anomalous quantum diffusion phenomenon is observed, where a larger $W$ results in a slower decay rate under strong disorder. Periodic oscillations of quantum diffusion are always present in boundary disordered systems. Research indicates that these periodic oscillations are caused by interlayer quantum jumps and result from the degenerate energy spectra between layers, independent of the system’s shape, size, boundary conditions, or lattice energy scale, and are applicable to different atomic structures. In disordered systems, the differences in interlayer energy spectra lead to the decay of periodic oscillations, and both the decay rate and frequency can be adjusted through the disorder strength $W$ and the interlayer coupling strength $u$ .