論文の公開元へMagneto-optical and magnetic properties of two-dimensional materials
概要
The Faraday (Kerr) effect is a magneto-optical phenomenon in which polarization direction of a linearly-polarized incident light is rotated upon transmission through (reflection by) a material, in the presence of an external magnetic field parallel to the direction of propagation of the incident light. In a three-dimensional (3D) material, the magnitude of the Faraday rotation is proportional to thickness of the material and strength of the magnetic field. The origin of the Faraday effect in a 3D material is non-identical refraction indexes for left-handed and right-handed circularly polarized lights (which constitute a linearly-polarized light) when the magnetic field is applied. Because the reflection of light depends on the refractive index, the Kerr rotation is also generated. The discovery of graphene in 2004 have drawn researchers to investigate the Faraday and the Kerr effects in thin films and two-dimensional (2D) materials. A thin-film or a 2D material possesses a relatively large carrier density even though the thickness of the material is negligibly small (in the order of 10−6 10−9 m). In the 2D materials, the magnetic field generates the optical Hall conductivity which is proportional to the charge density, and gives rise to the Faraday and the Kerr rotations. The Faraday and the Kerr effects without magnetic field have been observed in a thin film topological insulator as a result of spin-orbit coupling and intrinsic magnetization. In particular, the spin-orbit coupling and intrinsic magnetization gives rise to the quantum anomalous Hall (QAH) state, where the Hall conductivity is quantized. However, a general description of the Faraday and the Kerr effects in 2D materials is not yet discussed. A general description of a 2D material in the QAH state is given by the Haldane model, which has been experimentally realized by using cold atoms in an optical lattice. Moreover, it is theoretically predicted that the Haldane model can be synthesized in the form of Fe-based ferromagnetic insulators in a honeycomb lattice, in which electrons in occupied bands are fully polarized in one spin direction owing to the strong Hund coupling in Fe. Therefore, it is meaningful to investigate the Faraday and the Kerr effects in the Haldane model, which is the first subject of this thesis.
The second subject of this thesis is orbital magnetization of graphene and the related 2D Dirac materials. It is known that undoped graphene possesses large dia- magnetism, because in the presence of an external magnetic field, states of massless of electrons at the valence bands coalesce to the zeroth Landau level. As a result, the free energy of graphene increases with increasing magnetic field, which gives the orbital diamagnetism. Analytical formulas for orbital susceptibilities of graphene re- lated materials, including monolayer transition-metal dichalcogenides and the Weyl semimetals, can be obtained by applying the Euler-Maclaurin formula in the calcula- tion of thermodynamic potential. However, calculation of orbital magnetization with the Euler-Maclaurin formula yields a divergent result. It is because we need to con- sider an infinite number of the Landau levels in the valence bands in the expression of the thermodynamic potential. Moreover, an experimental measurement of the orbital magnetization of graphene has been performed for wide ranges of magnetic field and temperature. Therefore, a method to derive analytical expression of magnetization of graphene is required to identify the origin of the magnetization behaviour for a given magnetic field and temperature.
The purposes of the thesis are: (1) to investigate the Faraday and the Kerr rotations in 2D materials without an external magnetic field in the Haldane model, and (2) to formulate analytical expressions for orbital magnetizations of graphene and related 2D materials, and thus to explain the origin of the dependences of magnetizations on the magnetic field and temperature.
In Chapter 2, we discuss calculation methods for the first purpose. Here, we derive energy dispersion of the Haldane model and optical conductivities of 2D material by using the linear response theory. The analytical formulas of absorption probability, as well as the angles of the Faraday and the Kerr rotations are derived by solving the Maxwell equations with boundary conditions at the 2D material.
In Chapter 3, we discuss calculation methods for the second purpose, in which we derive the Landau levels and thermodynamic potential of the 2D materials.
In Chapter 4, we derive analytical expressions for longitudinal and the Hall con- ductivities of the Haldane model, which are the origins of the Faraday and the Kerr rotations. Maximum Faraday and Kerr rotations are generated when the photon en- ergy matches the energy band gap, due to the singularity in the real part of the Hall conductivity. Our treatment on the Faraday and the Kerr rotations is relevant to determine the topological phases in 2D materials. Moreover, our analytical formulas for optical conductivities can be applied to explain optical absorption of circularly- polarized lights in various 2D materials, such as silicene and monolayer transition- metal dichalcogenides.
In Chapter 5, we derive analytical expressions for orbital magnetizations of the 2D Dirac materials by using the zeta function to regularize infinite summation of the Landau levels. Our formula reproduces empirical fitting for orbital magnetization of undoped graphene in strong field/low temperature and weak field/high temperature limits. In the case of heavy Dirac fermions, we show that the magnetization is robust with respect to temperature and impurity scattering. Further, we demonstrate that the opening of band gap in the 2D materials can be detected from decreasing amplitude of the de Haas-van Alphen (dHvA) effect. Our results reproduce the experimental results without fitting procedure.
