Honeycomb layered oxides are a novel class of nanostructured materials comprising alkali or alkaline earth metals intercalated into transition metal slabs. The intricate honeycomb architecture and layered framework endows this family of oxides with a tessellation of features such as exquisite electrochemistry, unique topology and fascinating electromagnetic phenomena. Despite having innumerable functionalities, these materials remain highly underutilized as their underlying atomistic mechanisms are vastly unexplored. Therefore, in a bid to provide a more in-depth perspective, we propose an idealised diffusion model of the charged alkali cations (such as lithium, sodium or potassium) in the two-dimensional (2D) honeycomb layers within the three-dimensional (3D) crystal of honeycomb layered oxide frameworks. This model not only explains the correlation between the excitation of cationic vacancies (by applied electromagnetic fields) and the Gaussian curvature deformation of the 2D surface, but also takes into consideration, the quantum properties of the cations and their inter-layer mixing through quantum tunnelling. Through this work, we offer a novel theoretical framework for the study of 3D layered materials with 2D cationic diffusion currents, as well as providing pedagogical insights into the role of topological phase transitions in these materials in relation to Brownian motion and quantum geometry.

蜂窝状层状氧化物是一类新型的纳米结构材料，包括嵌入过渡金属板中的碱金属或碱土金属。错综复杂的蜂窝状结构和分层框架赋予该系列氧化物以细分的特征，例如精美的电化学，独特的拓扑结构和引人入胜的电磁现象。尽管具有无数的功能，但由于尚未充分探究其潜在的原子机制，因此这些材料仍未得到充分利用。因此，为了提供更深入的了解，我们提出了带电碱金属阳离子（例如锂，钠或钾）在三维（2D）蜂窝层内的理想扩散模型3D）蜂窝状分层氧化物骨架的晶体。该模型不仅说明了阳离子空位的激发（通过施加的电磁场）与二维表面的高斯曲率变形之间的相关性，而且还考虑了阳离子的量子性质及其通过量子隧穿进行的层间混合。通过这项工作，我们为研究具有2D阳离子扩散电流的3D层状材料提供了一个新颖的理论框架，并提供了有关拓扑相变在这些材料中与布朗运动和量子几何有关的作用的教学见识。阳离子的量子性质及其通过量子隧穿的层间混合。通过这项工作，我们为研究具有2D阳离子扩散电流的3D层状材料提供了一个新颖的理论框架，并提供了有关拓扑相变在这些材料中与布朗运动和量子几何有关的作用的教学见识。阳离子的量子性质及其通过量子隧穿的层间混合。通过这项工作，我们为研究具有2D阳离子扩散电流的3D层状材料提供了一个新颖的理论框架，并提供了有关拓扑相变在这些材料中与布朗运动和量子几何有关的作用的教学见识。