Local Magnetic Moments and Hyperfine Fields at Ta Impurities Diluted in X Fe 2 ( X = Y , Gd , Yb ) Laves Phases Compounds

Neste trabalho, estudamos, à temperatura finita, a sistemática na formação dos momentos locais de uma impureza de Ta diluída em compostos intermetálicos de fases de Laves XFe 2 (X = Y, Gd, Yb). Utilizamos um hamiltoniano de Hubbard, extendido à duas sub-redes acopladas, para descrever as matrizes de fases de Laves. A interação eletrônica d-d é tratada via método da integral funcional, na aproximação quase estática. Efeitos da pressão dependentes da temperatura estão incluídos, considerando a interação elétron-fônon induzida que renormaliza a hibridização do elétron puro. Os campos hiperfinos magnéticos calculados, relacionados aos momentos magnéticos locais, estão em bom acordo com os dados experimentais avaliados.


Introduction
The Laves phase intermetallic compounds XFe 2 crystallize either in the cubic C15 or hexagonal C14 structure. 1They exhibit an interesting variety of behaviors related to the changes in their magnetic, electronic, and lattice structures.For instance, in the intermetallic compound YFe 2 the magnetic order is sustained mainly by the itinerant electrons of the Fe sublattice and a small magnetic moment is induced at the Y sublattice.On the other hand the compounds of GdFe 2 and YbFe 2 exhibit magnetization associated both with the localized spins of the rare earth sublattice and the itinerant electrons of Fe sublattice.Although the rare earth Laves phase intermetallic compounds have been extensively studied, many properties of them still remain open.Therefore, the investigation of the magnetic hyperfine fields in the Laves phases intermetallic compounds may provide valuable information on the magnitude of the itinerant (3d) and also, in the case of rare earth intermetallic compounds, the localized (4f) magnetic moments.In this work we study, at finite temperature, the formation of the local magnetic moment and the systematics of the magnetic hyperfine fields at a Ta impurity diluted on the X site of the Laves phase intermetallic compounds XFe 2 (X = Y, Gd, Yb).Experimental data show that Ta impurity diluted in the XFe 2 intermetallic hosts enters in the X site.

Theoretical Model
In order to calculate the magnetic properties of the XFe 2 intermetallic hosts, we use a two sublattice Hamiltonian, describing a subsystem of itinerant d-electrons coupled with localized f-spins of rare earth ions. 4One has: (1) where x 0 ( Fe 0 ) is the energy of the center of X (Fe) sublattice, d j (d j ) is the creation (annihilation) operator, n j = d j d j is the number operator.T lj XX (T lj

FeFe
) is the hopping integral between only X (Fe) sub-lattice atoms, and T XFe is the corresponding to the processes involving X and Fe sublattices atoms.J df is an exchange interaction parameter and U Fe (U X ) is the Coulomb interaction parameter for the Fe (X).Notice that in the case of the YFe 2 host the last term of the equation (1) does not appear, since there is no magnetic moment at the Y sublattice.
5][6][7] In this framework of the functional integral method, the initial Hamiltonian with Coulomb correlations is mapped into an e ective one-body Hamiltonian in which the electrons are under the action of fluctuating charge ( ) and spin ( ) fields.These fluctuating fields which are randomly distributed all over the sites, define a "disorder problem" in both sublattices.We treat this intrinsic disorder in the Coherent Potential Approximation (CPA) and so one defines e ective media Fe and X to restore the translational invariance of the pure Laves phase host.The self-energies are self consistently determined by a CPA equation, 7 and thereby the X sublattice host is completely described at any finite temperature.
We consider that the Ta impurity diluted in the X sublattice, defines a Wolf-Clogston problem in the e ective medium at the X sublattice.The local d-density of states per spin direction at the impurity site is given by (2)   where G X 0 X 0 (z) is the local perturbed Green function given by: (3) and g X 0 X 0 (z) is the host Green function written in terms of the self-energy.The energy Ta 0 should be self consistently determined by the Friedel screening condition (4) where N X is the change in the occupation number at the X sublattice calculated by: 8 (5) The electron occupation number at the impurity site (n 0 ) is obtained by integrating the local density of states up to the Fermi level F , i.e., (6)   f( ) being the Fermi function.The local d-magnetic moment at the Ta impurity is then given by (7)   We also consider the s-p sublattice band to describe hyperfine interactions. 9The local s-p magnetic moment is obtained by (8)   where m X d is the X sublattice d-magnetization and the parameter is of order of 0.1. 9The total magnetic hyperfine field at the Ta impurity site, is made up by a conduction electron polarization (CEP) contribution due to the s-p conduction electrons and by a core polarization (CP) due to the d conduction electrons.One has (9)   where A(Z imp ) is the Fermi-Segrè contact coupling parameter and A d cp is d-core polarization parameter. 10

Results and Discussions
In order to obtain the numerical results for a Ta impurity diluted at XFe 2 metallic host, we need to follow some selfconsistent calculation steps.
Firstly, we describe the intermetallic hosts in the absence of the impurity.Hence we solve two set of equation: (i) for the itinerant system (d-electrons), and (ii) for the localized system due to the 4f rare earth electrons, in the cases of GdFe 2 and YbFe 2 hosts.So, the electronic and magnetic properties of the system are described properly.Notice that, as mentioned before the s electrons are not considered explicitly and they contribute to the appearance of the local moment through equation (8).Moreover, since the s-p band is large and flat, the local s-moment is constant with temperature, and is not considerately affected by the presence of the Ta impurity since the charge screening is made by the d-band.
Secondly, once the self-consistent calculations for the pure host were made at 0 K temperature, for a fixed set of the model parameters, we determine the self-energies Fe and X for at a given temperature.After that, we determine  the impurity energy Ta 0 via the Friedel screening condition and calculate the local magnetic moment and the related magnetic hyperfine fields at a Ta impurity.
In Figures 1(b), 2(b), and 3(b) we plot as a function of temperature, the calculated local magnetic moments at a Ta impurity dilute in YFe 2 , GdFe 2 , and YbFe 2 .
In Figures 1(a), 2(a), and 3(a) we plot as a function of temperature, the calculated magnetic hyperfine field at a Ta impurity dilute in YFe 2 , GdFe 2 , and YbFe 2 .Our theoretical calculations exhibit a very good agreement with experiments, 2,3 which show that the magnetic hyperfine Vol. 19, No. 2, 2008   field at the Ta impurity in GdFe 2 and YbFe 2 is antiparallel to the magnetization of the Fe sublattice being positive in both cases.
In Figure 4, we show a semi phenomenological calculation to include pressure effects. 11In this case, we assume that external pressure excites the elastic degrees of freedom of the host.Therefore one has an electronphonon interaction which modifies the pure electronic hybridization-like term, 13,14 V X = Ta -X ( F ) (see equation ( 5)).When the electron-phonon interaction is included in equation ( 5), one has a temperature dependence of V .Figure 4 illustrates such pressure effects.The experimental data were obtained for a high pressure of 7.7 GPa and our theoretical results are again in a good agreement with the experimental data reported in reference 3. So, we have shown, that a functional integral approach, using a mean field approximation can describe the magnetic hyperfine field behavior of Laves phase intermetallics.The theory should be enlarged to account also for light rare earth, in the beginning of the 4f series such as Ce, Pr and Nd, following the calculation made at T = 0 K made in reference 15.

Figure 1 .
Figure 1.(a) Magnetic hyperfine field at Ta impurity diluted in YFe 2 intermetallic host.The dotted line corresponds to the CEP contribution and the dashed line corresponds to the CP contribution.The solid line represents the total hyperfine The squares represent the experimental data collected from reference 2. (b) Local magnetic moments at Ta impurity diluted in YFe 2 intermetallic host.The dotted line corresponds to the m ~c(0) contribution and the dashed line corresponds to the m ~d(0) contribution.

Figure 2 .
Figure 2. (a) Magnetic hyperfine field at Ta impurity diluted in GdFe 2 intermetallic host.The dotted line corresponds to the CEP contribution and the dashed line corresponds to the CP contribution.The solid line represents the total hyperfine field.The circles represent the experimental data collected from reference 3. (b) Local magnetic moments at Ta impurity diluted in GdFe 2 intermetallic host.The dotted line corresponds to the m ~c(0) contribution and the dashed line corresponds to the m ~d(0) contribution.

Figure 3 .
Figure 3. (a) Magnetic hyperfine field at Ta impurity diluted in YbFe 2 intermetallic host.The corresponds to the CEP and the dashed line corresponds to the CP contribution.The solid line represents the total hyperfine field.The triangles represent the experimental data collected from reference 3. (b) Local magnetic moments at Ta impurity diluted in YbFe 2 intermetallic host.The dotted line corresponds to the m ~c(0) contribution and the dashed line corresponds to the m ~d(0) contribution.

Figure 4 .
Figure 4. Magnetic hyperfine field at Ta impurity diluted in GdFe 2 intermetallic host.The solid line corresponds to the high pressure calculated total magnetic hyperfine field and the dashed line to the total calculated hyperfine field at normal pressure.Triangles and circles represent experimental data collected from reference 3. Inset: Local magnetic moments at Ta impurity in GdFe 2 intermetallic host with pressure effect.Dotted line and dashed line represent the m ~c(0) and the m ~d(0) contributions, respectively.