This course applies basic quantum mechanics principles (Schrödinger equation and perturbation theory) to derive the band structure theories of crystalline solids and understand the multiphysics nature of materials (including electrical, optical, optoelectronic, and topological properties). Topics in this course include single-particle Schrodinger equation and its application in several quantum mechanical systems; Dirac notation, non-degenerate and degenerate perturbation theories and their applications in hydrogen and helium atoms; classical free-electron gas model, quantum free-electron theory, quantum density of states, Fermi-Dirac distribution, Maxwell Boltzmann Distribution, Fermi energy, and Fermi surface; Bloch’s theorem, approaching band model through Schrödinger equation; nearly free-electron model, tight-binding model, Kronig-Penney model for deriving the formation of discrete energy levels and band structures of crystalline solids; apply band structures to classify materials and understand electrical, optical, and topological properties of recently emerging materials systems (two-dimensional materials and topological insulators etc).
The course provides the fundamentals of crystallography, symmetry, diffraction and defects in materials, and explores the relationship between crystal symmetry and the directional properties of materials. Topics in this course include crystal structure, point groups, crystallographic restriction theorem, symmetry operations, and the use of symmetry in the tensor representation of crystal properties; Bragg’s law, Fourier analysis, reciprocal lattice, Brillouin zones, diffraction conditions, Laue equations, x-ray, electron and neutron diffraction, and structure determination; point defects (lattice vacancies and colour centres), dislocations (Burger vectors, stress fields, grain boundaries, and dislocation density, multiplication and slip). The course will deepen the understanding on how the macroscopic behaviours of materials, including electronic, optical, magnetic and mechanical properties of materials, originate from fundamental crystallography, symmetry, and microstructure.