Magnetic ground states of the tetragonal U2Fe2Sn-type structure using an anisotropic RKKY exchange modeling

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Abstract

In the light of numerous magnetic data obtained on the ternary compounds R2T2X (R=U, Ce; T=Ni, Rh, Pd and X=Sn, In), which crystallize in the tetragonal U2Fe2Sn-type structure (P4/mbm space group) but exhibit various magnetic structures, we demonstrate the efficiency of a Ruderman-Kittel-Kasuya-Yosida model based on a non-spherical Fermi surface to account for the magnetic interactions involved in these systems. Considering as magnetic ground states the basis vectors of the irreducible representations associated to P4/mbm space group, (4h) Wyckoff position [crystallographic site for Ce or U] and k=(000), k=(0012). k=(12120) and k=(121212) propagation vectors, we construct the ground state map as a function of the Fermi wave vector components kF and kFz. We show that the relative locations of the ground state domains in the (kF,kFz)-plane are coherent with the magnetic structures observed previously in (Ce1−xUx)2Pd2.05Sn0.95, U2Pd2+xSn1−x, and U2(Ni1−xPdx)2Sn solid solutions. More particularly, at 1.5 K the x-dependent sequence of magnetic structures in U2(Ni1−xPdx)2Sn system is clearly modeled: from (k=(0012),Γ10(φ=π)) for x=0 to (k=(000), NC1) for x=1 via (k=(121212),τ10) for x=0.30. In the (k=(121212),τ10) stability domain, the (k=(0012),Γ8) structure is calculated to be at a low energy gap above, which is consistent with its stabilization displayed at upper temperature around 8.0 K.

Introduction

During the last decade many investigations were devoted to the ternary compounds with the formula R2T2X where R=rare earth element, U or Np; T=transition element and X=In or Sn. Many of these compounds crystallize in the tetragonal U2Fe2Sn-type structure as U2T2X (T=Fe, Co, Ni, Ru, Rh, Pd) [1], [2], [3], Np2T2X (T=Co, Ni, Ru, Rh, Pd) [3], [4] and R2Pd2Sn (R=Ce, Nd, Gd, Tb, Dy, Ho and Er) [5]. Moreover, the compounds based on palladium show a range of homogeneity as U2Pd2+xSn1−x [6], Ce2Pd2+xSn1−x [7] or Ce2Pd2+xIn1−x [8] and adopt a structure deriving from U2Fe2Sn-type. Their interesting physical properties result from the competition between the intersite Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction and the onsite Kondo coupling: for instance in the sequence U2Fe2Sn→U2Co2Sn→U2Ni2Sn, the uranium atom exhibits respectively Pauli paramagnetism, spin fluctuation and antiferromagnetism behavior [2], [9].

Neutron diffraction studies were reported on many of these compounds confirming a variety of magnetic structures. Table 1 summarizes the studies performed on the stannides and indides based on uranium or cerium (label of the structure is described on Fig. 1). Excepted U2(Ni0.70Pd0.30)2Sn stannide, all the magnetic structures are characterized by only one commensurate wave vector k=(000) or k=(0012).

The numerous magnetic data have lead us to investigate a modeling of the magnetic properties of this family of ternary compounds. In reference to the studies dedicated to the solid solutions U2(Ni1−xPdx)2Sn [18] and (Ce1−xUx)2Pd2.05Sn0.95 [10], it has been shown that a RKKY model based on an ellipsoidal instead of a spherical Fermi surface, consistently accounts for the magnetic interactions between (U, Ce) localized moments in these two systems. Considering as magnetic ground states the basis vectors for irreducible representations for P4/mbm space group (U2Fe2Sn-type) with (U, Ce) magnetic moments in (4h) Wyckoff positions and the two propagation vectors k=(000) and k=(0012), realistic (TN,TC versus x) phase diagrams have been calculated for both solid solutions in the frame of mean-field approximation for order-disorder temperature determination. More precisely, the magnetic crossover from ferromagnetic to antiferromagnetic structures involved in the (Ce1−xUx)2Pd2.05Sn0.95 system at x=0.3 [10], has been also predicted at a close x-value. Concerning the U2(Ni1−xPdx)2Sn system [18], the occurrence of a minimum in TN=f(x) curve, giving the composition dependence of the Néel temperature, at x=0.30–0.35 has been reproduced and demonstrated to be associated to the change in magnetic structure, from the U2Ni2Sn one (k=(0012)) [14] to the U2Pd2Sn one (k=(000) [6], [11]). In order to prove this magnetic transition, neutron powder diffraction have been performed for two selected compounds U2(Ni0.70Pd0.30)2Sn [17] and U2(Ni0.55Pd0.45)2Sn [19]. The latter adopts below TN=20.5(5)K the same magnetic structure as U2Pd2Sn whereas the first one exhibits two magnetic transition temperatures: TN=11.5(5)K associated to the paramagnetic →(k=(0012),Γ8) transition, and TN′=8.0(5)K below which three magnetic structures have been observed, (k=(0012),Γ8,Γ2), and (k=(121212),τ10). They are supposed to be associated to different volumes of the sample, where the main volume fraction corresponds to the (k=(121212),τ10) structure [17]. In our previous modeling the k=(121212) propagation vector has not been taken into account. It has to be done now, and it is the aim of the present paper. The study of the Fermi wave vector dependence of the ground states characterized by the propagation vector k=(000) and k=(0012) is extended to the new ground states characterized by the propagation vectors k=(121212) and k=(12120). The k=(12120) vector is introduced for homogeneity reasons with respect to the k=(000) vector. We hope that the extension of the RKKY modeling to new propagation vectors will explain the complex magnetic phase diagram of the U2(Ni1−xPdx)2Sn near the x=0.30 critical composition.

Section snippets

Magnetic structures

The ground state map is calculated considering the irreducible magnetic configurations (basis vectors) resulting from magnetic group theory applied to P4/mbm space group: (4h) Wyckoff position (the crystallographic sites of the atoms supporting the magnetic moments of compounds listed in Table 1) and the four propagation vectors k=(000), k=(0012), k=(12120) and k=(121212). The basis configurations for the propagation vectors k=(000) and k=(0012) have been determined in Refs. [6], [11], [14] (

Energetics

The magnetic energy of the system is written as a classical Heisenberg potential E:E=−i,jJijSi·Sj,where Jij is the exchange integral between the spins Si and Sj of two atoms in the (4h) site of the P4/mbm space group. The coupling between Si and Sj is modeled within the indirect RKKY mechanism [22], [23], [24]. In the framework of a free-electron conduction and an uniaxial anisotropy of the electron mass in the band; mx=my=mmz where z is parallel to the tetragonal c-axis, it was shown [18]

Energy degeneracy of magnetic structures

The scalar form of the exchange integrals J(kF,Rij) leads to the energy degeneracy of several magnetic structures previously defined for the four propagation vectors k=(000), k=(0012), k=(12120) and k=(121212) (Fig. 1a and b). This is the case, for example, of the NC3 (or Γ1,ρ4,τ4) configuration which can be deduced from the NC2 (or Γ3,ρ2,τ2) structure by a π/2-rotation of the spins. C3 and the Γ10 configurations, characterized by a φ-angle of disorientation of magnetic moments in the (a, b

Conclusion

The main purpose of that study was to explain, in the framework of an anisotropic RKKY exchange modeling, the complex magnetic phase diagram of the U2(Ni1−xPdx)2Sn system [6], [14], [17], [18]. The first result is that the (k=(121212),τ10) magnetic structure, experimentally displayed below TN′=8.0(5)K on the U2(Ni0.70Pd0.30)2Sn compound, is able to be stabilized for specific Fermi wave vector components among the whole ground states characterized by the four propagation vectors k=(0,0,0), k=(001

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