First-principles calculation of dynamical properties of insulators in finite electric fields and anomalous Hall conductivity of ferromagnets based on Berry phase approach

We present first-principles methods for calculating two distinct types of physical quantities within the framework of density functional theory: the response properties of an insulator to finite electric fields, and the anomalous Hall conductivity of a ferromagnet. Both of the methods are closely related to the same ingredient, namely the Berry phase, a geometric phase acquired by a quantum system transporting in parameter space. We develop gauge-invariant formulations in which the random phases of Bloch functions produced by numerical subroutines are irrelevant. First, we provide linear-response methods for calculating phonon frequencies, Born effective charge tensors and dielectric tensors for insulators in the presence of a finite electric field. The starting point is a variational total-energy functional with a field-coupling term that represents the effect of the electric field. This total-energy functional is expanded with respect to both small atomic displacements and electric fields within the framework of density-functional perturbation theory. The linear responses of field-polarized Bloch functions to atomic displacements and electric fields are obtained by minimizing the second-order derivatives of the total-energy functional. The desired second-order tensors are then constructed from these optimized first-order field-polarized Bloch functions. Next, an efficient first-principles approach for computing the anomalous Hall conductivity is described. The intrinsic anomalous Hall conductivity in ferromagnets depends on subtle spin-orbit-induced effects in the electronic structure, and recent {it ab-initio} studies found that it was necessary to sample the Brillouin zone at millions of k-points to converge the calculation. We start out by performing a conventional electronic-structure calculation including spin-orbit coupling on a uniform and relatively coarse k-point mesh. From the resulting Bloch states, maximally localized Wannier functions are constructed which reproduce the {it ab-initio} states up to the Fermi level. With inexpensive Fourier and unitary transformations the quantities of interest are interpolated onto a dense k-point mesh and used to evaluate the anomalous Hall conductivity as a Brillouin-zone integral. The present scheme, which also avoids the cumbersome summation over all unoccupied states in the Kubo formula, is applied to bcc Fe, giving excellent agreement with conventional, less efficient first-principles calculations. Finally, we consider another {it ab-initio} approach for computing the anomalous Hall conductivity based on Haldane's Fermi-surface formulation. Working in the Wannier representation, the Brillouin zone is sampled on a large number of equally spaced parallel slices oriented normal to the total magnetization. On each slice, we find the intersections of the Fermi surface sheets with theslice by standard contour methods, organize these into a set of closed loops, and compute the Berry phase of the Bloch states as they are transported around these loops. The anomalous Hall conductivity is proportional to the sum of the Berry phases of all the loops on all the slices.Ph.D.Includes bibliographical references (p. 134-138)