Local magnetic moments and hyperfine fields at Ta impurities diluted in XFe2 (X = Y, Gd, Yb) Laves phases compounds

ARTICLE

Local magnetic moments and hyperfine fields at Ta impurities diluted in XFe₂ (X = Y, Gd, Yb) Laves phases compounds

Amós Troper^{I, II,*}; Nilson A. de Oliveira^II; Marcus V. Tovar Costa^III; Alexandre L. de Oliveira^IV

^ICentro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud, 150, 22290-180 Rio de Janeiro-RJ, Brazil

^IIInstituto de Física, Universidade do Estado do Rio de Janeiro, Rua São Francisco Xavier, 524, 20550-013 Rio de Janeiro-RJ, Brazil

^IIIInstituto de Aplicação, Universidade do Estado do Rio de Janeiro, Rua Santa Alexandrina, 288, 20261-232 Rio de Janeiro-RJ, Brazil

^IVCentro Federal de Educação Tecnológica de Química de Nilópolis / RJ, Rua Lúcio Tavares, 1045, 26530-060 Nilópolis-RJ, Brazil

ABSTRACT

In this work, we study the systematics, at finite temperature, of the formation of local magnetic moments at a Ta impurity diluted in intermetallic Laves phases compounds XFe₂ (X = Y, Gd, Yb). We use an extended two-coupled sublattice Hubbard Hamiltonian, to describe the Laves phases host. The d-d electronic interaction is treated via a functional integral approach in the quasi-static saddle point approximation. Temperature dependent pressure effects are included considering induced electron-phonon interaction which renormalizes the pure electron hybridization. The calculated magnetic hyperfine fields related to the obtained local magnetic moments, are in a quite good agreement with available experimental data.

Keywords: intermetallic compounds, magnetic hyperfine fields, impurities

RESUMO

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 XFe₂ (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 XFe₂ crystallize either in the cubic C15 or hexagonal C14 structure.¹ They exhibit an interesting variety of behaviors related to the changes in their magnetic, electronic, and lattice structures. For instance, in the intermetallic compound YFe₂ 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 GdFe₂ and YbFe₂ 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 XFe₂ (X = Y, Gd, Yb). Experimental data show that Ta impurity diluted in the XFe₂ intermetallic hosts enters in the X site.^2,3

Theoretical Model

In order to calculate the magnetic properties of the XFe₂ intermetallic hosts, we use a two sublattice Hamiltonian, describing a subsystem of itinerant d-electrons coupled with localized f-spins of rare earth ions.⁴ One has:

where is the energy of the center of X (Fe) sub-lattice, is the creation (annihilation) operator, is the number operator. is the hopping integral between only X (Fe) sub-lattice atoms, and T^XFe is the corresponding to the processes involving X and Fe sub-lattices 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 YFe₂ host the last term of the equation (1) does not appear, since there is no magnetic moment at the Y sublattice.

We use the functional integral technique to treat the d-d electron-electron interaction.^4-7 In this framework of the functional integral method, the initial Hamiltonian with Coulomb correlations is mapped into an effective one-body Hamiltonian in which the electrons are under the action of fluctuating charge (n) and spin (x) 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 effective media and 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 effective medium at the X sublattice. The local d-density of states per spin direction at the impurity site is given by

where is the local perturbed Green function given by:

and is the host Green function written in terms of the self-energy. The energy should be self consistently determined by the Friedel screening condition

where is the change in the occupation number at the X sublattice calculated by:⁸

The electron occupation number at the impurity site (n_0s) is obtained by integrating the local density of states up to the Fermi level e_F , i.e.,

f(e) being the Fermi function. The local d-magnetic moment at the Ta impurity is then given by

We also consider the s-p sublattice band to describe hyperfine interactions.⁹ The local s-p magnetic moment is obtained by

where is the X sublattice d-magnetization and the parameter g is of order of 0.1.⁹ The 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

where A(Z_imp) is the Fermi-Segrè contact coupling parameter and is d-core polarization parameter.¹⁰

Results and Discussions

In order to obtain the numerical results for a Ta impurity diluted at XFe₂ metallic host, we need to follow some self-consistent 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 GdFe₂ and YbFe₂ 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 and for at a given temperature. After that, we determine the impurity energy 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 YFe₂, GdFe₂, and YbFe₂.

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 YFe₂, GdFe₂, and YbFe₂. Our theoretical calculations exhibit a very good agreement with experiments,^2,3 which show that the magnetic hyperfine field at the Ta impurity in GdFe₂ and YbFe₂ 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.¹¹ In this case, we assume that external pressure excites the elastic degrees of freedom of the host. Therefore one has an electron-phonon interaction which modifies the pure electronic hybridization-like term,^13,14 (see equation (5)).

When the electron-phonon interaction is included in equation (5), one has a temperature dependence of V_s. 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.

Acknowledgment

This article was written for a special volume in honor of Professor Ricardo Ferreira. Professor Ricardo Ferreira is a true "natural philosopher" and his contribution to the development of natural sciences (physics, chemistry and biology) in Brazil has been extremely important. In his scientific work, originality and imagination are always present, thus being a source of inspiration to a whole generation of scientists.

Received: August 21, 2007

Published on the web: February 29, 2008

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Publication Dates

• Publication in this collection
  08 Apr 2008
• Date of issue
  2008

History

• Accepted
  29 Feb 2008
• Received
  21 Aug 2007