School of Science Department of Physics 56 Theoretical Study of Properties of Real Materials Supervisor: LIU Junwei / PHYS Student: NG Man Tik / PHYS Course: UROP 1100, Summer This report provides a bridge for beginners with an introductory understanding of quantum mechanics to grasp the tight-binding model. It begins by introducing crystals as lattices with discrete translational and rotational symmetries, which leads to Bloch’s Theorem and the reciprocal lattice. The empty lattice approximation, computed with degenerate perturbation theory, is discussed to illustrate the periodic energy property, the First Brillouin Zone, and the emergence of band gaps. Then, approximation techniques like the Linear Combination of Atomic Orbitals and the Variational Principle are discussed for solving a complicated Hamiltonian and introducing the occurrence of bonding. Finally, the tight-binding method is explained with the square lattice as an example. Theoretical Study of Properties of Real Materials Supervisor: LIU Junwei / PHYS Student: NG Max Ching To / PHYS-IRE Course: UROP 1100, Summer This article employed the method of tight binding model to obtain the band structures of infinite graphene, graphene nanoribbon with zigzag edge, and graphene nanoribbon with armchair edges. It is shown that infinite graphene behaves like a semiconductor and zigzag graphene nanoribbon behaves like a conductor. Armchair graphene nanoribbon behaves either like a semiconductor or an insulator depending on size. Axion Cosmology Supervisor: LIU Tao / PHYS Student: WONG Po Leung / PHYS-IRE Course: UROP 1100, Summer Dark matter is one component in the ΛCDM model, a model that is by far the simplest and can provide adequate explanations for many of the observed features of our universe. The triumph of ΛCDM model makes dark matter well accepted by cosmologists, despite its hypothetical and undetermined nature. Until now, the identity of dark matter is still waiting to be unravelled. Many models have been proposed in past decades as candidates for dark matter. Among these, axion is one of the most popular possibilities. In this report, we aim at presenting the features of dark matter and axion respectively, and discuss how axion can be a candidate of dark matter.
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