Recent progress in wave packet dynamics based on the insight of Berry pertaining to adiabatic evolution of quantum systems has led to the need for a new property of a Bloch state, the Berry curvature, to be calculated from first principles. We report here on the response to this challenge by the ab initio community during the past decade. First we give a tutorial introduction of the conceptual developments we mentioned above. Then we describe four methodologies which have been developed for first-principle calculations of the Berry curvature. Finally, to illustrate the significance of the new developments, we report some results of calculations of interesting physical properties such as the anomalous and spin Hall conductivity as well as the anomalous Nernst conductivity and discuss the influence of the Berry curvature on the de Haas-van Alphen oscillation.

We present an ab~initio description of the spin Hall effect in metals. Our approach is based on density functional theory in the framework of a fully relativistic Korringa-Kohn-Rostoker method and the solution of a linearized Boltzmann equation including the scattering-in term (vertex corrections). The skew scattering mechanism at substitutional impurities is considered. Spin-orbit coupling in the host as well as at the impurity atom and the influence of spin-flip processes are fully taken into account. A sign change of the spin Hall effect in Cu and Au hosts is obtained as a function of the impurity atom, and even light elements like Li can cause a strong effect. It is shown that the gigantic spin Hall effect in Au can be caused by skew scattering at C and N impurities which are typical contaminations in a vacuum chamber.

On the basis of detailed ab initio studies, the influence of strain on the anisotropy of the transport distribution of the thermoelectrics Bi2Te3 and Sb2Te3 was investigated. Both tellurides were studied in their own as well as in their copartner's lattice structure to gain insight into the electrical transport in epitaxial heterostructures composed of both materials. It is shown that the anisotropy of the transport distribution overestimates the experimental findings for Bi2Te3, implying anisotropic scattering effects. An increase of the in-plane lattice constant leads to an enhancement of the transport anisotropy for p doping, whereas the opposite occurs for n doping. The recent findings and special features of the transport distribution are discussed in detail in relation to the topology of the band structures.

The Fermi surface that characterizes the electronic band structure of crystalline solids can be difficult to image experimentally in a way that reveals local variations. We show that Fermi surfaces can be imaged in real space with a low-temperature scanning tunneling microscope when subsurface point scatterers are present: in this case, cobalt impurities under a copper surface. Even the very simple Fermi surface of copper causes strongly anisotropic propagation characteristics of bulk electrons that are confined in beamlike paths on the nanoscale. The induced charge density oscillations on the nearby surface can be used for mapping buried defects and interfaces and some of their properties.

Ab initio calculations are presented for the giant magnetoresistance (GMR) of Fe/Cr multilayers. The electronic structure of Fe/Cr superlattices is calculated within the spin density functional theory. The spin-dependent bulk impurity scattering is treated in a fully quantum mechanical way using a Green function formalism. The transport is described quasiclassically. The calculations suggest that the GMR in thin multilayers is mainly determined by the electronic structure of the layered system. Spin-dependent impurity scattering can reduce, increase, or invert the effect.