The Division of Materials Research and the Office of Cyberinfrastrcture contribute funds to this award. It supports theoretical research and education on the electronic properties of novel materials in which orbital currents play an important role. The objectives are (i) to develop the formal theory of such systems, making use of mathematical concepts from differential geometry; (ii) to develop accurate, efficient, robust and informative algorithms for computing the associated properties of materials; and (iii) to apply these methods to study actual and as-yet unsynthesized materials, especially ones having potential technological applications. A major thrust of the research program is to make further developments in the theory of the electronic structure of materials in which time-reversal symmetry is broken, for example ferromagnets, and in the theory of topological insulators. Mathematical approaches related to Berry phases and the Wannier representation will be utilized to investigate these more general problems. These techniques have proven useful for understanding electric polarization, orbital magnetization, and the anomalous Hall conductivity. Theoretical investigations will be carried out to better understand two classes of topological insulators. The first is the theoretically simpler but experimentally more elusive "Chern" or "quantum anomalous Hall" insulator, of which no known experimental realizations exist to date some 20 years after a theory describing them appeared. This work should clarify the expected physical properties of such materials and may suggest further avenues for experimental searches. The second are the "Z2 topological insulators," several examples of which have been discovered in the last five years. A second major thrust of the research program concerns the calculation of the linear magnetoelectric couplings of crystalline insulators. Methods for such calculations are still in their infancy, but are ripe for further development. Using first-principles methods, all of the various contributions to the magnetoelectric coupling will be calculated, including purely electronic, lattice-displacement-mediated, and strain-mediated ones, for several prototypical materials. This project is expected to lead to fundamental advances in the understanding of the electronic structure of materials with unusual magnetic or topological order, and to contribute to the development of novel materials that are promising for commercial applications, especially ones involving the coupling of electrical and magnetic responses. This project will also contribute to developing formal theory and methods to enable first principles calculations of the properties of these materials.
NON-TECHNICAL SUMMARY The Division of Materials Research and the Office of Cyberinfrastrcture contribute funds to this award. It supports research and education in computational condensed-matter theory, with a focus on obtaining a deeper understanding of novel materials in which orbital currents play an essential role. Magnetic phenomena generally fall into two classes: those explained by a quantum mechanical property of the electron known as spin, and those related to the presence of microscopic currents that flow at the atomic scale. The effects of these latter "orbital currents" are sometimes secondary. For ordinary magnets such as iron, they account for less than 10% of the magnetism. However, in recent years there has been an outpouring of interest in certain novel materials for which the orbital currents play the dominant role. In a series of remarkable developments a few years ago, for example, theoretical predictions of "topological insulators" were quickly followed by experimental confirmations. By definition, electric currents cannot flow in the interior of an insulator, but a topological insulator has the unusual property that there are guaranteed to be current-carrying channels at the surfaces. Essentially, the "topological" organization of the electrons in the bulk enforces a certain corresponding organization of the atomic-scale orbital currents so as to produce a net current at the surface. Such phenomena could have important practical applications; one example might be materials that can convert electrical impulses to magnetic impulses and vice versa. The PI's program is focused on obtaining a detailed understanding of these unusual materials and their magnetoelectric phenomena, with activities spanning from formal theory, development and implementation of new computer algorithms, predictive computer simulations, and pedagogical dissemination of the results.
Overview. This project involved research and education in computational condensed-matter theory, with a focus on obtaining a deeper understanding of novel materials in which orbital currents play an essential role. Magnetic phenomena generally fall into two classes: those explained by electron spin, and those related to the presence of microscopic currents that flow at the atomic scale. The effects of these latter "orbital currents" are sometimes secondary, as for ordinary magnets such as iron in which they account for less than 10% of the magnetism. However, in recent years there has been an outpouring of interest in certain novel materials for which the orbital currents play the dominant role. In a series of remarkable developments a few years ago, for example, theoretical predictions of so-called "topological insulators" were quickly followed by experimental confirmations. By definition, electric currents cannot flow in the interior of an insulator, but a topological insulator has the unusual feature that there are guaranteed to be current-carrying channels at the surfaces. Essentially, the "topological" organization of the electrons in the bulk enforces a certain corresponding organization of the atomic-scale orbital currents so as to produce a net current at the surface. Such phenomena may have important practical applications, e.g., for the development of materials that convert electrical to magnetic impulses and vice versa. The project outcomes are summarized as follows. Methods development. The usual method of treating electrons in crystalline solids is to work in terms of electron states labeled by a "crystal momentum" describing the speed and direction of motion in the crystal. An alternative viewpoint is that of the "Wannier representation" where instead one works with localized wave packets of electrons. The two are mathematically equivalent, but one or the other may be more practical or insightful under different circumstances. We have shown that a "hybrid Wannier representation," formulated in terms of electron states characterized by momentum in two dimensions and by localized position in the third dimension, provides an especially useful and insightful method of distinguishing between normal and topological insulators, identifying the type of topological insulator, and picking out the region of momentum responsible for the topological behavior. Code packages embodying the algorithms necessary for this analysis have been made publicly available. Topological insulators. One type of magnetic topological insulator that may be of special interest is the two-dimensional "quantum anomalous Hall" or "Chern" insulator. Such a state would behave like the quantum Hall effect, for which the Nobel Prize was given in 1985, but without the need for any external magnetic field or ultra-low temperature. The possibility of such a state was first proposed in 1988, but was demonstrated experimentally for the first time in 2013, but still at very low temperatures. In a series of papers, we have theoretically proposed possible ways preparing surfaces of magnetic insulators in such a way as to realize the Chern insulator state at much higher temperature, stimulating experimental efforts to realize our proposals. Another class of topological insulator s are the non-magnetic ones typified by Bi2Se3, which has been the subject of intense experimental development. Using first-principles calculations, we explained how and why the properties of this material change when alloyed, or built into layered structures, with non-topological constituents. Magnetoelectric coupling. The linear magnetoelectric coupling (MEC) is a property that some magnetic materials have, and describes a cross-coupling between electric and magnetic fields in the material. The MEC can be divided into "lattice" contributions coming from atomic displacements in response to the field, and "electronic" ones that would result even if the atoms were frozen in place. They can also be divided into "spin" and "orbital" contributions in the sense of the first paragraph above. For the first time for any material, we have systematically computed all four kinds of contributions and compared their strengths and characters, using Cr2O3 as a prototypical material. The orbital-electronic contribution is particularly tricky, and we have developed methods allowing for its practical computation for the first time. Broader impacts. Under this project, we have developed several computer code packages, mainly for analyzing the topological properties of electrons in crystalline materials, and have made these packages publicly available to the scientific community. The research has resulted in the training, mentorship, and career development of students and postdocs. Applications of the research may eventually contribute to technological advances with commercial impact.
