Topology, Connectivity, and Charge Transport in Gallium Nitride: A Graph Theory Framework

Abstract: In this work, we explore the electrical conductivity of Gallium Nitride(GaN) using graph theory to model the connectivity and charge transport pathways within its molecular and crystalline structure. By representing the compound as a graph, where atoms are nodes and chemical bonds are edges, we analyze the network’s conductivity using concepts such as adjacency matrices, Laplacian matrices, and percolation thresholds. The adjacency matrix (A) encodes the connectivity between atoms, where each element Aij equals 1 if there is a bond between nodes i and j, and 0 otherwise. The Laplacian matrix(L), defined as L = D−AL (where D is the degree matrix that represents the number of bonds for each node), captures the topological structure of the system and is used to calculate the eigenvalues, which represent the vibrational modes or energy states within the material. The smallest non-zero eigenvalue, known as the spectral gap, plays a crucial role in determining electrical conductivity, as it is inversely proportional to the material’s ability to conduct charge. The conductivity of GaN is influenced by the network’s topology, including factors such as bond strength, coordination number, and the presence of conduction paths. The percolation threshold the critical connectivity point at which the network transitions from insulating to conducting is also an essential factor in determining the material’s electrical properties. This graph-theoretic approach provides a novel framework for understanding the conductive properties of metal nitride systems, offering insights into how molecular structure and network topology govern charge transport. It can also be extended to other compounds for optimizing their conductivity in various applications. Keywords: Graph theory, Electrical conductivity, Gallium Nitride, Network analysis, Percolation theory.