Effective Local Moment Research Articles

Mixing of t2g−eg orbitals in 4d and 5d transition metal oxides

Using exact diagonalization, we study the spin-orbit coupling and interaction-induced mixing between $t_{2g}$ and $e_g$ $d$-orbital states in a cubic crystalline environment, as commonly occurs in transition metal oxides. We make a direct comparison with the widely used $t_{2g}$ only or $e_g$ only model, depending on electronic filling. We consider all electron fillings of the $d$-shell and compute the total magnetic moment, the spin, the occupancy of each orbital, and the effective spin-orbit coupling strength (renormalized through interaction effects) in terms of the bare interaction parameters, spin-orbit coupling, and crystal field splitting, focusing on the parameter ranges relevant to 4d and 5d transition metal oxides. In various limits we provide perturbative results consistent with our numerical calculations. We find that the $t_{2g}$-$e_g$ mixing can be large, with up to 20\% occupation of orbitals that are nominally "empty", which has experimental implications for the interpretation of the branching ratio in experiments, and can impact the effective local moment Hamiltonian used to study magnetic phases and magnetic excitations in transition metal oxides. Our results can aid the theoretical interpretation of experiments on these materials, which often fall in a regime of intermediate coupling with respect to electron-electron interactions.

Magnetic fluctuations and effective magnetic moments inγ-iron due to electronic structure peculiarities

Applying the local density and dynamical mean field approximations to paramagnetic \gamma-iron we revisit the problem of theoretical description of magnetic properties in a wide temperature range. We show that contrary to \alpha-iron, the frequency dependence of the electronic self-energy has a quasiparticle form for both, t_{2g} and e_g states. In the temperature range T=1200-1500 K, where \gamma-iron exist in nature, this substance can be nevertheless characterized by temperature-dependent effective local moments, which yield relatively narrow peaks in the real part of the local magnetic susceptibility. At the same time, at low temperatures \gamma-iron (which is realized in precipitates) is better described in terms of itinerant picture. In particular, the nesting features of the Fermi surfaces yield maximum of the static magnetic susceptibility at the incommensurate wave vector q_{max} belonging the direction q_X-q_W (q_X=(2\pi/a)(1,0,0),q_W=(2\pi/a)(1,1/2,0), a is a lattice parameter) in agreement with the experimental data. This state is found however to compete closely with the states characterized by magnetic wave vectors along the directions q_X-q_L-q_K, where q_L=(2\pi/a)(1/2,1/2,1/2), q_K=(2\pi/a)(3/4,3/4,0). From the analysis of the uniform magnetic susceptibility we find that contrary to \alpha-iron, the Curie-Weiss law is not fulfilled in a broad temperature range, although the inverse susceptibility is nearly linear in the moderate-temperature region (1200-1500 K). The non-linearity of the inverse uniform magnetic susceptibility in a broader temperature range is due to the density of states peak located close to the Fermi level. The effective exchange integrals in the paramagnetic phase are estimated on the base of momentum dependent susceptibility.

Effect of local charge fluctuations on spin dynamics inLa2CuO4

We explore the effect of local charge fluctuations on the spin response of a Mott insulator by deriving an effective spin model, and studying it using Schwinger boson mean field theory. Applying this to ${\mathrm{La}}_{2}\mathrm{Cu}{\mathrm{O}}_{4}$, we show that an accurate fit to the magnon dispersion relation, measured by Coldea et al. [Phys. Rev. Lett. 86, 5377 (2001)] is obtained with Hubbard model parameters $U\ensuremath{\approx}2.34\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$, and $t\ensuremath{\approx}360\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$. These parameters lead to estimates of the staggered magnetization $({m}_{s}\ensuremath{\approx}0.25)$, spin wave velocity $(c\ensuremath{\approx}800\phantom{\rule{0.3em}{0ex}}\mathrm{meV}\phantom{\rule{0.2em}{0ex}}\mathrm{\AA{}})$, and spin stiffness $({\ensuremath{\rho}}_{s}\ensuremath{\approx}24\phantom{\rule{0.3em}{0ex}}\mathrm{meV})$. In particular the staggered magnetization as well as the effective local moment are renormalized to smaller values compared to the Heisenberg model due to local charge fluctuations in the Hubbard model. The dynamical structure factor shows considerable weight in the continuum along the zone boundary as well as secondary peaks that may be observed in high resolution neutron scattering experiments.

Magnetovolume and magnetoelastic effects in dilute Cr-Ga alloys

The thermal expansion and elastic constants of a series of dilute Cr-Ga alloys are reported between 77 and 500K. In addition to magnetic anomalies at the Neel temperature, TN, well defined anomalies in both properties have been observed for the first time at the incommensurate-commensurate transition temperatures of this system which has a triple point near 0.5 at.% Ga and 280K where the paramagnetic, incommensurate (I) and commensurate (C) spin density wave (SDW) states coexist. The temperature dependence of the magnetovolume, Delta omega , of alloys showing I-C transitions is analysed to yield the ratio (m2(0))CSDW/(m2(0))ISDW=1.5+or-0.2, where (m2(0)) is the mean squared value of the effective local moment at 0K. Using this result the magnetic contribution to the bulk modulus at 0K in a Cr+1.00 at.% Ga alloy is found to be mod Delta B(0) mod CSDW=(0.20+0.05)*1011 N m-2. Also for this alloy, for example, delta ln TN/ delta omega ISDW=20+or-4, in good agreement with independent measurements. For some alloys the calculated pressure dependence of the Neel temperatures agrees fairly well with direct measurements. Above TN the magnetovolume due to antiferromagnetic spin fluctuations decays roughly exponentially with temperature.

Correlation effects in ferromagnetism of transition metals

The effect of electronic correlations on the ferromagnetic instabilities, effective local moments, and fluctuations in spin and charge in a transition metal is discussed with a model Hamiltonian which includes intraatomic Coulomb and exchange interactions. The one-particle properties of $d$ electrons are described by bcc and fcc canonical bands. We find that reduction of density fluctuations and formation of local moments change the Stoner criterion and the condition for the ground state to be completely ferromagnetic. The magnetic states of Fe, Co, and Ni are studied in more detail, where the electrons are found to be considerably correlated for realistic model parameters. Redistribution of electrons between the ${e}_{g}$ and ${t}_{2g}$ states seen there causes deviations from the Stoner theory and is responsible for different exchange splittings of both types of states. The results are in qualitative agreement with the experimental data and suggest the use of anisotropic exchange correlation potentials in transition metals.

Thermodynamic properties of magnetic impurities in metals

Our previously developed trial function scheme has been generalized to finite temperatures in order to investigate the thermodynamic properties of the nondegenerate Wolff model. Using a “microcanonical ensemble ansatz”, the change of free energy due to the introduction of a single impurity to the host metal has been derived. The impurity contributions to the thermodynamic quantities can then be calculated. We found a broad maximum for the specific heat. The magnetic susceptibility is temperature-insensitive at lower T, and turns into the Curie-Weiss behavior at higher T. The effective local moment vanishes at T = 0, while with rising T it increases to a saturated value of reduced moment. Such characteristic features agree qualitatively with the experimental findings.

Variational treatment of electronic correlations ind-band metals

We discuss the effects of electronic correlation on the ground state energy, effective local moment and charge fluctuations for a model of transition metals. The model consists of a bcc or fcc-canonicald-band with intraatomic Coulomb and exchange interaction. We find that for realistic interaction parameters the correlated state shows enhanced spinfluctuations indicating the formation of local moments and strongly reduced charge fluctuations compared to the HF state. We also find that in generale g andt 2g -states can show a different extent of correlation which suggests the use of anisotropic exchange correlation potentials in transition metals. The application of our model to bcc iron and fcc nickel is also considered.

Summary of Mössbauer evidence associated with the Kondo effect

M\"ossbauer-effect measurements provided the first direct evidence of the reduction of individual effective localized moments on magnetic impurities with decreasing temperature. While other techniques such as magnetic susceptibility measure static collective properties, the M\"ossbauer and nuclear-magnetic-resonance techniques enjoy the capability of directly measuring the spin polarization within atomic dimensions of a given nucleus on a time scale of $\ensuremath{\tau}\ensuremath{\sim}{10}^{\ensuremath{-}7}\ensuremath{-}{10}^{\ensuremath{-}9}$ sec. Though the theoretical prediction of a reduction in the effective local moment for a simple and somewhat unrealistic Hamiltonian was made by Kondo some eight years ago and the theory reformulated, refined and improved, only phenomenological models have been developed to reduce the M\"ossbauer-effect data. In this note we analyze data for dilute Fe concentrations in several cubic transition metals for evidence of the reduction of the moments localized about Fe impurities. These results and others from the literature will be summarized and compared.

New Solution for Exchange Scattering in Dilute Alloys

The $s\ensuremath{-}d$ exchange model is treated using equations of motion truncated at the lowest nontrivial order, following Nagaoka. The coupled equations are reduced to a single nonlinear integral equation for the conduction-electron $t$ matrix, which depends only on energy. An approximation to the integral operator which treats the Kondo divergence accurately permits this equation to be transformed to a differential equation which is exactly integrable. The solution agrees with the leading terms of perturbation calculations above the Kondo critical temperature ${T}_{K}$, and passes through this temperature smoothly, reaching the unitarity limit at zero temperature. A different analytic continuation of the $t$ matrix is trivially found which acquires nonphysical singularities below ${T}_{K}$. At low temperatures this form is shown to be identical to Abrikosov's solution and to Suhl's solution prior to analytic continuation. The resistivity of dilute alloys is calculated. Noninteracting impurities are shown to give no contribution to the specific heat. The effective local moment entering the magnetic susceptibility is found to be almost completely canceled at zero temperature for spin-\textonehalf{} impurities.