We present a review of experimental and theoretical studies of the anomalous Hall effect (AHE), focusing on recent developments that have provided a more complete framework for understanding this subtle phenomenon and have, in many instances, replaced controversy by clarity. Synergy between experimental and theoretical work, both playing a crucial role, has been at the heart of these advances. On the theoretical front, the adoption of Berry-phase concepts has established a link between the AHE and the topological nature of the Hall currents which originate from spin-orbit coupling. On the experimental front, new experimental studies of the AHE in transition metals, transition-metal oxides, spinels, pyrochlores, and metallic dilute magnetic semiconductors, have more clearly established systematic trends. These two developments in concert with first-principles electronic structure calculations, strongly favor the dominance of an intrinsic Berry-phase-related AHE mechanism in metallic ferromagnets with moderate conductivity. The intrinsic AHE can be expressed in terms of Berry-phase curvatures and it is therefore an intrinsic quantum mechanical property of a perfect cyrstal. An extrinsic mechanism, skew scattering from disorder, tends to dominate the AHE in highly conductive ferromagnets. We review the full modern semiclassical treatment of the AHE together with the more rigorous quantum-mechanical treatments based on the Kubo and Keldysh formalisms, taking into account multiband effects, and demonstrate the equivalence of all three linear response theories in the metallic regime. Finally we discuss outstanding issues and avenues for future investigation.

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Anomalous hall effect

Multiferroics, defined for those multifunctional materials in which two or more kinds of fundamental ferroicities coexist, have become one of the hottest topics of condensed matter physics and materials science in recent years. The coexistence of several order parameters in multiferroics brings out novel physical phenomena and offers possibilities for new device functions. The revival of research activities on multiferroics is evidenced by some novel discoveries and concepts, both experimentally and theoretically. In this review, we outline some of the progressive milestones in this stimulating field, especially for those single-phase multiferroics where magnetism and ferroelectricity coexist. First, we highlight the physical concepts of multiferroicity and the current challenges to integrate the magnetism and ferroelectricity into a single-phase system. Subsequently, we summarize various strategies used to combine the two types of order. Special attention is paid to three novel mechanisms for multiferroicity generation: (1) the ferroelectricity induced by the spin orders such as spiral and E-phase antiferromagnetic spin orders, which break the spatial inversion symmetry; (2) the ferroelectricity originating from the charge-ordered states; and (3) the ferrotoroidic system. Then, we address the elementary excitations such as electromagnons, and the application potentials of multiferroics. Finally, open questions and future research opportunities are proposed.

Multiferroicity: the coupling between magnetic and polarization orders

Within the past 20 years or so, there has occurred an explosion of interest in the magnetic behavior of pyrochlore oxides of the type $A_{2}^{3+}$$B_{2}^{4+}$O$_{7}$ where $A$ is a rare-earth ion and $B$ is usually a transition metal. Both the $A$ and $B$ sites form a network of corner-sharing tetrahedra which is the quintessential framework for a geometrically frustrated magnet. In these systems the natural tendency to form long range ordered ground states in accord with the Third Law is frustrated, resulting in some novel short range ordered alternatives such as spin glasses, spin ices and spin liquids and much new physics. This article attempts to review the myriad of properties found in pyrochlore oxides, mainly from a materials perspective, but with an appropriate theoretical context.

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Magnetic Pyrochlore Oxides

Resonant Inelastic X-ray Scattering (RIXS)
 is an X-ray in, X-ray out technique that enables one to study the 
dispersion of excitations in solids. In this thesis, we investigated how
 various elementary excitations of transition metal oxides show up in 
RIXS spectra.

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Resonant Inelastic X-ray Scattering Studies of Elementary Excitations

Materials with a coexistence of magnetic and ferroelectric order — multiferroics — provide an efficient route for the control of magnetism by electric fields. The study of multiferroics dates back to the 1950s, but in recent years, key discoveries in theory, synthesis and characterization techniques have led to a new surge of interest in these materials. Different mechanisms, such as lone-pair, geometric, charge-ordering and spin-driven effects, can support multiferroicity. The general focus of the field is now shifting into neighbouring research areas, as we discuss in this Review. Multiferroic thin-film heterostructures, device architectures, and domain and interface effects are explored. The violation of spatial and inversion symmetry in multiferroic materials is a key feature because it determines their properties. Other aspects, such as the non-equilibrium dynamics of multiferroics, are underrated and should be included in the topics that will define the future of the field. Multiferroic materials exhibit magnetic and ferroelectric order at the same time and provide a way to control magnetism with electric fields. We discuss the mechanisms supporting multiferroicity, multiferroic thin films and heterostructures, the non-equilibrium dynamics of multiferroics, fundamental symmetry issues and the impact of multiferroics on other research areas.

The evolution of multiferroics

We investigate the various cation substitutional dopants in Ga2O3 for the possibility of p–type conductivity using density functional theory. Our calculations include both standard density functional theory and hybrid functional calculations. We demonstrate that all the investigated dopants result in deep acceptor levels, not able to contribute to the p–type conductivity of Ga2O3. In light of these results, we compare our findings with other wide bandgap oxides and reexamine previous experiments on zinc doping in Ga2O3.

On the feasibility of p-type Ga2O3

The optical magnetoelectric (ME) effect, i.e., the change of optical absorption upon the reversal of the light propagation direction, has been investigated for a polar ferrimagnet GaFeO3. For dipole- and spin-forbidden d-d transition bands located at 1.2-2.3 eV, a clear signal of the optical ME effect (Deltaalphat approximately 3x10(-3)) is observed with an applied magnetic field as low as 500 Oe and a sample thickness (t) of 50 microm. The observation of a large ME effect in the present compound suggests a possible route to magnification of this novel phenomenon for application.

Optical magnetoelectric effect in the polar GaFeO3 ferrimagnet.

We report inelastic x-ray scattering studies of charge excitations in insulating Nd{sub 2}CuO {sub 4} as a function of incident photon energy. An excitation of {approximately}6 eV is observed when the incident photon energy is tuned through the Cu K edge. This is interpreted as resonantly enhanced inelastic x-ray scattering. Numerical calculations identify the 6 eV feature as a charge-transfer excitation to the antibonding state and suggest that nonlocal effects play a role in determining the shape of the resonance. {copyright} {ital 1998} {ital The American Physical Society}

Resonant Inelastic X-Ray Scattering in Nd{sub 2} CuO{sub 4}

We employ the nudged elastic band and the dimer methods within the standard density functional theory (DFT) formalism to study the migration of the oxygen and gallium vacancies in the monoclinic structure of $\ensuremath{\beta}\ensuremath{-}{\mathrm{Ga}}_{2}{\mathrm{O}}_{3}$ We identify all the first nearest neighbor paths and calculate the migration barriers for the diffusion of the oxygen and gallium vacancies We also identify the metastable sites of the gallium vacancies which are critical for the diffusion of the gallium atoms The migration barriers for the diffusion of the gallium vacancies are lower than the migration barriers for oxygen vacancies by 1 eV on average, suggesting that the gallium vacancies are mobile at lower temperatures Using the calculated migration barriers we estimate the annealing temperature of these defects within the harmonic transition state theory formalism, finding excellent agreement with the observed experimental annealing temperatures Finally, we suggest the existence of percolation paths which enable the migration of the species without utilizing all the migration paths of the crystal

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Migration mechanisms and diffusion barriers of vacancies in Ga 2 O 3

Various forms of carbon based complexes in GaN are studied with first-principles calculations employing Heyd-Scuseria-Ernzerhof hybrid functionals within the framework of the density functional theory. We consider carbon complexes made of the combinations of single impurities, i.e., CN−CGa, CI−CN, and CI−CGa, where CN, CGa, and CI denote C substituting nitrogen, C substituting gallium, and interstitial C, respectively, and of neighboring gallium/nitrogen vacancies ( VGa/ VN), i.e., CN−VGa and CGa−VN. Formation energies are computed for all these configurations with different charge states after full geometry optimizations. From our calculated formation energies, thermodynamic transition levels are evaluated, which are related to the thermal activation energies observed in experimental techniques such as deep level transient spectroscopy. Furthermore, the lattice relaxation energies (Franck-Condon shift) are computed to obtain optical activation energies, which are observed in experimental techniques such ...

A first-principles study of carbon-related energy levels in GaN. I. Complexes formed by substitutional/interstitial carbons and gallium/nitrogen vacancies

Masahiko Matsubara

Papers

On the feasibility of p-type Ga2O3

Optical magnetoelectric effect in the polar GaFeO3 ferrimagnet.

Resonant Inelastic X-Ray Scattering in Nd{sub 2} CuO{sub 4}

Migration mechanisms and diffusion barriers of vacancies in Ga 2 O 3

A first-principles study of carbon-related energy levels in GaN. I. Complexes formed by substitutional/interstitial carbons and gallium/nitrogen vacancies