Superexchange theory of electronic polarization driven by relativistic spin-orbit interaction at the half-filling

By applying Berry-phase theory for the effective half-filled Hubbard model, we derive an analytical expression for the electronic polarization driven by the relativistic spin-orbit (SO) coupling. The model itself is constructed in the Wannier basis, using the input from the first-principles electronic structure calculations in the local-density approximation, and then treated in the spirit of the superexchange theory. The obtained polarization has the following form: ${\bf P}_{ij} = \boldsymbol{\epsilon}_{ji} \boldsymbol{\cal P}_{ij} \cdot [\boldsymbol{e}_i \times \boldsymbol{e}_j]$, where $\boldsymbol{\epsilon}_{ji}$ is the direction of the bond $\langle ij \rangle$, $\boldsymbol{e}_i$ and $\boldsymbol{e}_j$ are the directions of spins in this bond, and $\boldsymbol{\cal P}_{ij}$ is the pseudovector containing all the information about the crystallographic symmetry of the considered system. The expression describes the ferroelectric activity in various magnets with noncollinear but otherwise nonpolar magnetic structures, which would yield no polarization without SO interaction, including the magnetoelectric (ME) effect, caused by the ferromagnetic canting of spins in the external magnetic field, and spin-spiral multiferroics. The abilities of this theory are demonstrated for the the analysis of linear ME effect in Cr$_2$O$_3$ and BiFeO$_3$ and properties multiferroic MnWO$_4$ and $\beta$-MnO$_2$. In all considered examples, the theory perfectly describes the symmetry properties of the induced polarization. However, in some cases, the values of this polarization are underestimated, suggesting that other effects, besides the spin and electronic ones, can also play an important role.Comment: 31 pages, 10 figure

Magnetic structure of noncentrosymmetric perovskites PbVO3 and BiCoO3

It is well known that if a crystal structure has no inversion symmetry, it may allow for Dzyaloshinskii-Moriya magnetic interactions, operating between different crystallographic unit cells, which in turn should lead to the formation of long-periodic spin-spiral structures. Such a behavior is anticipated for two simple perovskites PbVO3 and BiCoO3, crystallizing in the noncentrosymmetric tetragonal P4mm structure. Nevertheless, we argue that in reality PbVO3 and BiCoO3 should behave very differently. Due to the fundamental Kramers degeneracy for the odd-electron systems, PbVO3 has no single-ion anisotropy. Therefore, the ground state of PbVO3 will be indeed the spin spiral with the period of about one hundred unit cells. However, the even-electron BiCoO3 has a large single-ion anisotropy, which locks this system in the collinear easy-axis C-type antiferromagnetic ground state. Our theoretical analysis is based on the low-energy model, derived from the first-principles electronic structure calculations.Comment: 16 pages, 7 figures, 3 table

Magnetization induced local electric dipoles and multiferroic properties of Ba2CoGe2O7

Ba2CoGe2O7, crystallizing in the noncentrosymmetric but nonpolar structure, belongs to a special class of multiferroic materials, whose properties are predetermined by the rotoinversion symmetry. Unlike inversion, the rotoinversion symmetry can be easily destroyed by the magnetization. Moreover, due to specific structural pattern, the magnetic structure of Ba2CoGe2O7 is relatively soft. Altogether, this leads to the rich variety of multiferroic properties, where the magnetic structure can be easily deformed by the magnetic field, inducing the electric polarization in the direction, which depends on the direction of the magnetic field. In this paper, we show that all these properties can be successfully explained on the basis of realistic low-energy model, derived from the first-principles electronic structure calculations for the magnetically active Co 3d bands, and the Berry-phase theory of electric polarization. Particularly, we argue that the magnetization induced electric polarization in Ba2CoGe2O7 is essentially local and expressed via the expectation values of some dipole matrices, calculated in the Wannier basis of the model, and the site-diagonal density matrices of the magnetic Co sites. Thus, the basic aspects of the behavior of Ba2CoGe2O7 can be understood already in the atomic limit, where both magnetic anisotropy and magnetoelectric coupling are specified by density matrix. Then, the macroscopic polarization can be found as a superposition of electric dipoles of the individual Co sites. We discuss the behavior of interatomic magnetic interactions, main contributions to the magnetocrystalline anisotropy and the spin canting, as well as the similarities and differences of the proposed picture from the phenomenological model of spin-dependent p-d hybridization.Comment: 27 pages, 8 figure

Screening of Coulomb interactions in transition metals

We discuss different methods of calculation of the screened Coulomb interaction $U$ in transition metals and compare the constraint local-density approximation (LDA) with the GW approach. We clarify that they offer complementary methods of treating the screening and should serve for different purposes. In the GW method, the renormalization of bare on-site Coulomb interactions between 3d electrons occurs mainly through the screening by the same 3d electrons, treated in the random phase approximation (RPA). The basic difference of the constraint-LDA method is that it deals with the neutral processes, where the Coulomb interactions are additionally screened by the ``excited'' electron, since it continues to stay in the system. This is the main channel of screening by the itinerant ($4sp$) electrons, which is especially strong in the case of transition metals and missing in the GW approach, although the details of this screening may be affected by additional approximations, which typically supplement these two methods. The major drawback of the conventional constraint-LDA method is that it does not allow to treat the energy-dependence of $U$. We propose a promising approximation based on the combination of these two methods. First, we take into account the screening of Coulomb interactions in the 3d-electron-line bands located near the Fermi level by the states from the subspace being orthogonal to these bands, using the constraint-LDA methods. The obtained interactions are further renormalized within the bands near the Fermi level in RPA. This allows the energy-dependent screening by electrons near the Fermi level including the same 3d electrons.Comment: 25 pages, 5 figures, 2 table

Self-consistent linear response for the spin-orbit interaction related properties

In many cases, the relativistic spin-orbit (SO) interaction is regarded to be small and can be treated using perturbation theory. The major obstacle on this route comes from the fact that the SO interaction can also polarize the electron system and produce additional contributions to the perturbation theory, arising from the electron-electron interactions. In electronic structure calculations, it may even lead to necessity to abandon the perturbation theory and return to the self-consistently solution of Kohn-Sham-like equations with the effective potential $\hat{v}$, incorporating the effects of the electron-electron interactions and the SO coupling, even though the latter is small. In this work, we present the theory of self-consistent linear response (SCLR), which allows us to get rid of numerical self-consistency and formulate it analytically in the first order of the SO coupling. This strategy is applied to the Hartree-Fock solution of the effective Hubbard model, derived from electronic structure calculations in the Wannier basis. By using $\hat{v}$, obtained from SCLR, one can successfully reproduce results of ordinary calculations for the orbital magnetization and other properties in the first order of the SO coupling. Particularly, SCLR appears to be extremely useful for calculations of antisymmetric Dzyaloshinskii-Moriya (DM) interactions based on the magnetic force theorem. Furthermore, due to the powerful 2n+1 theorem, the SCLR theory allows us to calculate the magnetic anisotropy energy up to the third order of the SO coupling. The fruitfulness of this approach is illustrated on a number of example, including the spin canting in YTiO$_3$ and LaMnO$_3$, formation of spiral magnetic order in BiFeO$_3$, and the magnetic inversion symmetry breaking in BiMnO$_3$, which gives rise to both ferroelectric activity and DM interactions, responsible for the ferromagnetism.Comment: 42 pagse, 5 figure, 6 table

Lattice Distortion and Magnetism of 3d-t2gt_{2g} Perovskite Oxides

Several puzzling aspects of interplay of the experimental lattice distortion and the the magnetic properties of four narrow $t_{2g}$-band perovskite oxides (YTiO$_3$, LaTiO$_3$, YVO$_3$, and LaVO$_3$) are clarified using results of first-principles electronic structure calculations. First, we derive parameters of the effective Hubbard-type Hamiltonian for the isolated $t_{2g}$ bands using newly developed downfolding method for the kinetic-energy part and a hybrid approach, based on the combination of the random-phase approximation and the constraint local-density approximation, for the screened Coulomb interaction part. Then, we solve the obtained Hamiltonian using a number of techniques, including the mean-field Hartree-Fock (HF) approximation, the second-order perturbation theory for the correlation energy, and a variational superexchange theory. Even though the crystal-field splitting is not particularly large to quench the orbital degrees of freedom, the crystal distortion imposes a severe constraint on the form of the possible orbital states, which favor the formation of the experimentally observed magnetic structures in YTiO$_3$, YVO_, and LaVO$_3$ even at the HF level. Beyond the HF approximation, the correlations effects systematically improve the agreement with the experimental data. Using the same type of approximations we could not reproduce the correct magnetic ground state of LaTiO$_3$. However, we expect that the situation may change by systematically improving the level of approximations for dealing with the correlation effects.Comment: 30 pages, 17 figures, 8 tables, high-quality figures are available via e-mai