  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.
