| With rapid developments of spintronics, spin-orbit coupling (SOC) effect has attracted more and more attention. The spin-orbit coupling comes from relativistic effect and connects orbital momentum of an electron with its spin. Thus one can possibly manipulate the spin of an electron through the spin-orbit coupling without an external magnetic field, which may result in all-electrically controlled spin device. On the other hand, many exotic physical phenomena in solid materials result from spin-orbit coupling, for example, spin Hall effect, magnetocrystalline anisotropy and solid magneto-optic effect, etc. Recently, a new quantum state of matter, so-called topological insulator, occurs in some compounds involving heavy atoms, whose strong spin-orbit interactions can induce inverted band structures and many unique electronic structures.For those systems including heavy atoms, their electronic structures can usually be changed much by spin-orbit coupling, resulting in various physical properties. In this dissertation, we investigated the influence of spin-orbit coupling effect on the electronic structures of the clusters and oxides, containing transition-metal elements. There are seven chapters in this thesis.In the first chapter, we introduce the spin-orbit interaction and the related effect, i.e. magneto-optic effect, spin Hall effect and magnetocrystalline anisotropy. After that, the background of transition-metal clusters is briefly reviewed, and the recent progress made in the research about their spin-orbit coupling effect is also mentioned. At the end of this chapter, the basic theory and structural classification of transition-metal oxides are presented, and the recent research on oxides including iridium (Ir) and ruthenium (Ru) atoms is also summarized.In the second chapter, theoretical methods adopted in this thesis are introduced, including adiabatic approximation, density-functional theory and Bloch’s theorem, etc. The brief presentations to plane-wave method, pseudopotential approximation and projector augmented-wave method are also given. In certain systems, if the electron correlation effect is strong, the single electron approximation may not give reasonable description. Hubbard model was employed to deal with these systems, which is also introduced in this chapter.In the third chapter, we investigate the influence of spin-orbit coupling on electronic structures of TM@Au12and TM@Ag12(TM=3d,3d, and5d transition metal atoms) by using density functional theory. The following properties are included in our investigation, i.e. the dispersion of the frontier orbitals, symmetry, HOMO-LUMO gap, magnetic moments, and binding energies of systems. The spin-orbit coupling can generally disperse much the frontier orbitals of the clusters, especially for the clusters with heavy TM atoms in each series. Only the systems with closed-shells (TM@Au12, TM=Mo and W) or’closed-orbitals’(TM@Au12, TM=Ni, Pd, and Pt) obey the Ih2group after the spin-orbit coupling is considered, in which all the spin and orbital magnetic moments (total or local) are quenched to zero. The ’closed-orbitals’ are formed solely under strong mixing of the spin-up and-down states of the frontier orbitals. The HOMO-LUMO gaps and the spin and orbital magnetic moments in Au clusters are usually decreased by the spin-orbit coupling. The former can be ascribed to the dispersion of the frontier orbitals; while the latter, the increase of the hybridization between TM d and Au6s and5d states. The trend of the orbital magnetic moments in different clusters may be understood through superatom picture. The influence of spin-orbit coupling on the electronic structures of the Ag clusters is rather weaker than that of the Au clusters. For the last systems of each series (TM=Ni, Pd, Pt), there is large difference between Au and Ag clusters, i.e. the three TM@Ag12(TM=Ni, Pd, Pt) clusters do not form ’closed-orbitals’due to weak spin-orbit coupling effect.In the fourth chapter, the electronic structures of BaIrO3are studied. The on-site Coulomb interaction U is found to be important for the5d transition-metal oxides to obtain the magnetic ground state. Only within the LSDA+U+SOC scheme, correct electric states, which are consistent with experimental results, can be predicted for the compound. Most importantly, we find the system is a unique spin-orbital Mott insulator in triple Jeff=1/2states. The gap of the system is opened due to the splitting of upper and lower Hubbard bands (UHB and LHB) of the middle Jeff=1/2state. The multiplicity of Jeff=1/2states is determined by the special structural units of the Ir3O12trimers in the compound. Weak and non-collinear magnetic states are found for the system. Our work presents clearly how the lattice, spin, and orbital degrees of freedom compete to each other in BaIrO3.In the fifth chapter, we investigate the electronic structures of9R-BaRuO3and give the primary results. The total energy calculations including on-site Coulomb energy U suggest the ferrimagnetic state is the ground state of9R-BaRuO3, in which the middle Ru atom of Ru3O12trimer and its two neighbors are antiferromagnetic coupling. This result agrees well with the calculation of Heisenberg exchange constants.The spin-orbit coupling effect is also considered, and it produces little influence on electronic structures of this system. According to the Ru-4d partial density of states, the different roles of5orbitals(dz2, dxy, dx2-y2,dxz,dyz)of Ru-4d states for Ru-O and Ru-Ru interactions are discussed, which can help us understand the complicated interactions of this compound.In the sixth chapter, we consider the effect of strain (-7%to7%) on the electronic structures of double perovskite oxide Ba2MnWO6. The in-plane tensile strain can not remarkably change the electronic states of system. However, the transition from antiferromagnetic semiconductors to ferromagnetic metals apperas when the in-plane compressive strain exceeds5%. The mechanisms are analyzed.In the seventh chapter, a brief summary of the thesis is given, and some plans about the following work are presented.
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