Anomalous magnetic reordering in magnetodielectric terbium iron garnet at low temperatures

Anomalous magnetic reordering in magnetodielectric terbiumiron garnet at low temperatures

Mahieddine Lahoubi a,n, Bachir Ouladdiaf b

a Department of Physics, Laboratory L.P.S., Faculty of Sciences, Badji Mokhtar-Annaba University, PO Box -12, 23000 Annaba, Algeriab Institut Laue Langevin, BP 156, F-38042 Grenoble Cedex 9, France

a r t i c l e i n f o

Article history:Received 18 December 2013Received in revised form30 January 2014

Keywords:Terbium iron garnetFerrimagnetismNeutron diffractionRepresentation analysisRhombohedral subgroupDouble umbrella

a b s t r a c t

The paper deals with five topics: i) the single three-dimensional irreductible representation (Г4g¼T1g) ofthe paramagnetic space group Ia3d No. 230 is chosen according to the representation analysis of Bertautfor the interpretation of the neutron powder diffraction experiments performed on terbium iron garnet(Tb3Fe5O12); ii) the use of the method of the “symmetry lowering device” of Bertaut in order to select theappropriate rhombohedral subgroup of Ia3d which allows to deal with the case where the cubicdescription provides an incomplete answer to the changes observed below 160 K in the ferrimagneticstructure around the [1 1 1] axis from the Néel model toward the “double umbrella” observed at 13 K; iii)the magnetic modes belonging to the one-dimensional irreductible representation A2g of the highestrhombohedral subgroup R3c No. 167 are able to describe the occurrence of its anisotropic characterwhich steeply increases below 160 K due to the concomitant anisotropic effects; iv) the broad anomalyobserved near 54 K in the temperature dependences of the components of both sublattices of the Tb3þ

ions in the Wyckoff positions (6e) and (6e0) is explained partially on the basis of the concept of Belov ofthe strong paraprocess which has been termed “exchange-enhanced paramagnetism” at the so-called“low-temperature point” (TB); v) the results are related to the magnetodielectric effect in low magneticfield and to the significant coupling between exchange magnons and ligand-field excitations reportedrecently in this compound.

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1. Introduction

Rare earth iron garnets {RE3þ3 }[Fe3þ2 ](Fe3þ3 )O12 or REIG (whereRE3þ denotes a rare-earth ion or yttrium Y3þ) with eight formulaunits per unit cell of the cubic space group Ia3d ðO10

h ÞNo. 230discovered at Grenoble by Bertaut and Forrat [1] have attractedcontinual interest of many researchers due to their fundamentaland technological importance [2–4]. Due to the recent magnetodi-electric (MD) and magnetoelectric (ME) effects revealed in TbIG [5–7]these REIG are considered now as promising well known chemicalcompounds with advanced magnetic and dielectric properties.

The magnetic ions are distributed over three crystallographicsites of the paramagnetic cubic space group Ia3d: dodecahedral{24c} for the RE3þ , octahedral [16a] and tetrahedral (24d) for theFe3þ ions respectively. Their magnetic properties illustrate theNéel’ theory of ferrimagnetism [8] where the magnetic momentsof the RE3þ ions form a collinear ferrimagnetic sublattice anti-parallel to the resultant iron magnetization along the [1 1 1]

crystallographic axis below the Néel temperature (TN) which isapproximately the same for all REIG (55476 K) [9,10]. At lowtemperatures, the collinear model must be replaced by a non-collinear spin alignment of the RE3þ magnetic moments.

The first neutron powder diffractions (NPD) performed by Bertautet al., [11] and Tchéou et al., [12] have revealed a coaxial magneticstructure in terbium iron garnet Tb3Fe5O12 (or TbIG), the magneticmoments of the Tb3þ ions forming around the [1 1 1] axis, “oneumbrella” structure at 1.5 K [11,12]. Later, a “double umbrella”structure was found at 4.2 K by Lahoubi et al., [13] and 5 K by Hocket al., [14]. The first “double umbrella” seems to open below thecompensation temperature (Tcomp¼243.570.5 or 249.070.5 K forsingle crystal or powder sample respectively) [13] or below 130 K [15].

The large MD effects in TbIG reported on single crystal sampleat 2 K and in low external applied magnetic field (μ0 Ho0.2 T) areperceptible up to 150 K [5]. Kang et al., [6] have suggested in theirstudy of the far infrared transmission a possible connectionregarding to the significant coupling between exchange magnonsand ligand-field excitations of the Tb3þ ions occurring between 60and 80 K and the existence of two distinct behaviors above andbelow�140 K. However, a number of important questions about

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n Corresponding author.E-mail address: [email protected] (M. Lahoubi).

Please cite this article as: M. Lahoubi, B. Ouladdiaf, Journal of Magnetism and Magnetic Materials (2014), http://dx.doi.org/10.1016/j.jmmm.2014.02.015i

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the mechanisms that drive these low-field MD effects at themicroscopic and macroscopic levels remain unanswered.

The aim of this paper is to get a more clear understanding howthe “double umbrella” changes over a large temperature range inorder to determine in some way a possible connection with thelow-field MD effects. Our study is based on the symmetry analysisof Bertaut [16–20] in the interpretation of the observed NPDpatterns and may be considered as a continuation of the recentwork on TbIG where new refined parameters have been reportedat 5 K by one of us [21].

2. Experimental

The powder samples of TbIG prepared at the Institut Néel(Grenoble, France) by the standard method have been analyzedby X-ray diffraction (XRD) using a Siemens model D5000 diffract-ometer equipped with a diffracted-beam graphite monochromatorCuKα radiation (λKα1¼1.54056 Å) at room temperature. The selectedpowder sample of pure phase has any parasite phase with a latticeconstant ac¼12.43870.01 Ǻ in good agreement with the value(12.436 Ǻ) published previously in Ref. [22].

The NPD patterns have been collected firstly on the diffract-ometer D1B at the Institut Laue-Langevin (Grenoble, France). Thesample which is consisted of 3 cm3 of TbIG powder was held invanadium can of 4 mm diameter. The value of the wavelength λ isequal to 2.52 Å and only four temperatures (5, 13, 20 and 160 K)have been chosen. The registration time for a pattern varied from0.5 to 1.5 h and 2θ range up to 901 was covered. The resolution ofthe D1B multidetector is 0.21. At 13 and 160 K, the measurementswere recorded successively with the detector shifted by 0.11 inorder to improve the resolution. This procedure allows a betterdetermination of the intensity of the neutron powder diffractionlines especially the superstructure lines (hkl)n, as (1 1 0)n and(2 0 0)n which appear in low angles. The second set of NPDpatterns has been recorded on the diffractometer D2B with awavelength λ¼1.594 Å and a 2θ range from 5 to 1601 at thesame temperatures. The choice of this reflection excludes a λ/2contamination.

3. Results and discussion

3.1. Interpretation of the magnetic structures in the space group Ia3d

We use the usefulness four linear combinations of four spinvectors, one ferromagnetic mode labeled F and three antiferromag-neticmodes labeled G, C and Awhich form the “magnetic modes” ofthe irreductible representations (irreps) of the space group Ia3d inthe application of the representation analysis of Bertaut [16–20].The whole symmetry analysis has been performed by hand for onemagnetic sublattice of the RE3þ ions in the dodecahedral site {24c}(2 2 2) [11,12,23] and later by one of us [13,15,21,24,25] for threesublattices with the two iron ions Fe3þ in octahedral [16a](3) andtetrahedral (24d)(4) sites. In these ferrimagnetic materials, onlythe single three-dimensional irrep (Г4g¼T1g) among the ten irreps,Гig and Гiu with (i¼1–5) associated to the point group Oh of Ia3d isable to describe the magnetic structures contrary to the previousgroup theoretical results where a combination of two irreps hasbeen proposed, two three-dimensional irreps (Г4g¼T1g) and(Г5g¼T2g) [11,12] or a combination of three and one-dimensionalirreps [26]. If we take account the basis vectors of the three coupledmagnetic sublattices belonging to the even irreps Гig (i¼1–5) andthose for {24c}(2 2 2) and (24d)(4) belonging to the odd irreps Гiu

(i¼1–5) which have been gathered for the first time by one of us[24,25], it appears some compatibility criteria: all Гig and Гiu

with (i¼1–5) are active for {24c} but all Гiu are forbidden for[16a] and A1u and A2g forbidden for (24d). The basis vectors of thetwo-dimensional irrep Eg does not give rise to equal RE3þ

moments; those of the three-dimensional irrep (Г5g¼T2g) describethe RE3þ moments in the (1 1 1) plane and there are not compa-tible with the Néel model along the [1 1 1] axis. It can be easy toconfirm that this irrep which corresponds to the representationGM4þ in the Miller and Love notation accounts the ferromagneticcomponent along the [1 1 1] axis of the cubic crystal using theavailable BasIreps program [27]. We can conclude that only the irrep(Г4g¼T1g) must participate for any REIG to the solution of bothmagnetic structures (collinear and non-collinear) below TN inagreement with the arguments developed previously only for theRE3þ ions in the Wyckoff position {24c} for TbIG and HoIG [23]. Thefour magnetic modes labeled F, (ferromagnetic) G, C and A (anti-ferromagnetics) introduced by Bertaut [16] which form the “basis ofirreps” are used here in our cubic description.

At T¼13 K (Fig. 1a), two types of reflections are present in allthe observed NPD patterns: i) the reflections (hkl) where theobserved nuclear are mixed to the magnetic intensities whichresult from the collinear state of the Néel model along the easyaxis [1 1 1] below TN; ii) the pure magnetic reflections (hkl)n,forbidden by the nuclear space groupIa3d. Both reflections are

Fig. 1. Observed of the NPD (D1B) patterns of TbIG: (a) T¼13 K; (b) T¼160 K.

M. Lahoubi, B. Ouladdiaf / Journal of Magnetism and Magnetic Materials ∎ (∎∎∎∎) ∎∎∎–∎∎∎2






indexed in the chemical cell with the same general extinction ruleof the cubic space group Ia3d, (hþkþ l¼2n) with the propagationvector k¼0. Thus, the primitive translation of the crystallographiclattice (I), (1∣1/2,1/2,1/2) is then a primitive translation of themagnetic lattice (I). Taking into account the previous numberedpositions [21,24,25] of the Tb3þ ions in dodecahedral site {24c},we can conclude that the following spins Sj and Sjþ12 (j¼1–3) arecoupled ferromagnetically. Firstly, the pure magnetic lines (hkl)n,{(1 1 0)n, (3 1 0)n, (411, 330)n, (433, 530)n and (5 1 0)n} related tothe signature of the “double umbrella” of the Tb3þ moments, areobserved at 13 K (Fig. 1a).

Only (1 1 0)n presents a small intensity without ambiguity inthe observed NPD pattern at 160 K (Fig. 1b) with a good sensibilityof order to 0.3 % which is equal to the ratio of the peak tobackground normalized to the intensity of the reflection (2 1 1).

Secondly, the fact that the magnetic lines (2 2 2)n and (6 2 2)n

are absent in both NPD patterns leads to a major magneticsymmetry property: the spins vectors Sj and Sjþ6 (j¼1–3) of theTb3þ ions in site {24c} are coupled ferromagnetically and thesymmetry operation (1∣0,0,0) is an inversion center which impliesthat the modes Aj and Cj are absent and only the modes Fj and Gj

will be considered. From these experimental conditions, it appearsthat the Wyckoff positions {24c} of the Tb3þ ions split into sixmagnetically inequivalent sublattices, Cj and C0

j with (j¼1–3): ineach sublattice, we have four ions which are equivalent under bothsymmetry operations: (1∣0,0,0) and (1∣1/2,1/2,1/2), the sublatticesC2 and C3 are related to C1 by a rotation of 120 and 2401 around the[1 1 1] axis and also for C0

j.Thirdly, the pure magnetic lines (2 0 0)n and (600, 442)n which

have been observed previously at 5 K [14] and confirmed recentlyby one of us at the same temperature [21], are absent in bothpatterns as shown in Fig. 1a, b; this result induces the eliminationof the mode Fj and the associated magnetic structure factorFM{c}(hkl)n, will be written as a function of the mode Gj.

The refinement of the NPD pattern at 13 K within the singlethree-dimensional irrep (Г4g¼T1g) of Ia3d give rise to a “doubleumbrella” structure but with an absolute value of the magneticmoment of the sublattice C0

1 m01 above the free Tb3þ ion value

(9 μB). In fact, in the previous XRD measurements performed onpowder sample [28–30] and on mixed single crystals Tb3-xYx

Fe5O12 (x¼0, 0.46, 0.88, 1.35) [31] the rhombohedral distortionsobserved below 200 and 100 K respectively have been analyzed inthe space group R3c as the distorted crystallographic subgroup ofIa3d. At the opposite, it has been already shown from other XRDmeasurements carried out on pure TbIG single crystal [32] that thecrystallographic symmetry must be broken to rhombohedral spacegroupR3, another subgroup where the forbidden nuclear reflec-tions {2 2 2} which characterize the departure from the cubic tothe rhombohedral symmetry [33] do not exist from room tem-perature down to 95 K.

We must ask ourselves the question about the true highestrhombohedral subgroup ofIa3dthat one must consider during thedecreasing of the crystal symmetry order, in agreement with themagnetization measurements [34,35] which indicate that [1 1 1]remains the easy axis of magnetization in the whole temperaturerange? The application of the method of “symmetry loweringdevice” developed by Bertaut [20] is able to give a completeanswer to the above question and the refinements must be basednow on the rhombohedral symmetry.

3.2. Interpretation of the magnetic structures in the rhombohedralsubgroups of Ia3d

We choose the maximal non-isomorphic subgroups of type “I”proposed in 2010 by Wondratschek and Müller in the Interna-tional Tables of Crystallography [36].

3.2.1. Choice of the highest rhombohedral subgroups of Ia3dSome remarkable facts for the choice of the highest subgroups of

the space group Ia3d emerge here: I41/acdðD204hÞNo. 142 (tetragonal)

and R3cðD63dÞNo. 167 (rhombohedral). This second highest subgroup

is a rhombohedral space group with the following subgroups: threewith a rhombohedral symmetry {R32ðD7

3Þ,R3ðC23iÞ, R3cðC6

3vÞ} and onewith a monoclinic symmetry {C2/cðC6

2hÞ}. According to previous NPDstudy of the crystallographic and magnetic structures of TbIG at5 and 13 K described in the subgroup R3[14], the Tb3þ ions lieon general positions with site symmetry (1). Such description isvery general and one can obtain a manifold of possible Tb3þ spinconfigurations compatible with the 3-fold symmetry axis [1 1 1].The refined values of the components m1x and m0

1z have been foundabove the free ion value (9 μB) and a poor agreement has been foundbetween the calculated magnetizations MS

cal along the three maincrystallographic directions [1 1 1], [1 1 0] and [1 0 0] and the mag-netic measurements on single crystals MS

mag [24,25,34,35]. We areled to conclude that the true space subgroup of Ia3d at lowtemperatures is R3cinstead ofR3.

3.2.2. Magnetic structures in the subgroup R3c of Ia3dWe recall the correspondence between the sites of the mag-

netic ions in the two space groups Ia3d and R3c: {24c}(2 2 2) splitsin two sites (6e)(2) and (6e0)(2); (24d)(4) in one site (12f)(1) and[16a](3) in two sites (2b)(3) and (6d)(1). The three-dimensionalirrep (Г4g¼T1g) is reduced to (A2gþEg). The irrep Eg will not beconsidered. The possible magnetic modes which result from therepresentation analysis of Bertaut applied to the space group-R3chave been developed by one of us [21,24,25] and only thecorresponding magnetic structures belonging to the even one-dimensional irrep A2g are described here briefly.

We have found that it convenient to define the spins Sj and Sjþ6

(j¼1–3) of the RE3þ ion spins in the Wyckoff position (6e) asfollows:

Sj ¼ Sjþ6 ¼ Sj; cf þSj; ncf ¼ f n7agj; ðj¼ 1–3Þ ð1Þ

where the corresponding unit vectors gj, (j¼1–3) are related to theunit vectors {i, j, k} of the cubic axis respectively

g1 ¼1ffiffiffi

3p ð� iþ jþkÞ; g2 ¼

1ffiffiffi

3p ðþ i� jþkÞ; g3 ¼

1ffiffiffi

3p ðþiþ j�kÞ

ð2Þ

which are related to the unit vector n of the ternary axis [1 1 1]with the following relation

n¼ g1þg2þg3 ð3Þ

Here, the general solution is based on the following resultant basisvectors

Vcf ðA2gÞ ¼c11ðIÞþc11ðIIÞ; Vncf ðA2gÞ ¼ þ fc11ðIÞ�c11ðIIÞg ð4Þ

where Vcf(A2g)) and Vncf(A2g) are respectively the collinear andnon-collinear ferromagnetic modes whose combinations belong tothe same irreductible representation A2g. They give rise a non-collinear ferromagnetic structure around the ternary axis [1 1 1] ofthe RE3þ ion spins of the Wyckoff position (6e) (associated to thesublattice Cj, (j¼1–3) in the cubic notation). The spins Sj and Sjþ6

(j¼1–3) are situated in the three glide planes c, c.3 and c.32 ofR3c:each plane containing the ternary axis [1 1 1] (unit vector n) andone of the rhombohedral directions {[111], [111], [111]} repre-sented by the unit vectors gj, (j¼1–3). Consequently, one finds

c¼ ðn; g3Þ; c:3¼ ðn; g1Þ and c:32 ¼ ðn; g2Þ ð5Þ

For the RE3þ ion spins of the Wyckoff position (6e0) (associated tothe sublattices C0

j, (j¼1–3) in the cubic notation), the same model

M. Lahoubi, B. Ouladdiaf / Journal of Magnetism and Magnetic Materials ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3






constructed for (6e) can be used and the results are

Sj0 ¼ Sj0 þ6 ¼ Sj0 ; cf þSj0 ; ncf ¼ f 0n7a0gj ð6Þwith j¼1–3 and j0 ¼4–6. For a better comparison with the “cubicdescription” of the space group Ia3d, we choose a mixed coordinatesystem, i.e., the local axes {pj, qj, n} where the unit vectors pj (j¼1–3)associated to the low symmetry axes {[211], [121], [112]} are

p11ffiffiffi

6p ð�2iþ jþkÞ; p2 ¼

1ffiffiffi

6p ðþ i�2jþkÞ; p3 ¼

1ffiffiffi

6p ðiþ j�2kÞ ð7Þ

If one takes into account the reduced local axes {pj, n}, finally, wehave the following new equations

Sj ¼ Sjþ6 ¼ ðf 7a=3Þn7ð2ffiffiffi

2p

=3Þapj; ðj¼ 1–3Þ ð8Þ

Sj0 ¼ Sj0 þ6 ¼ ðf 07a0=3Þn7 ð2ffiffiffi

2p

=3Þa0pj ð9ÞAccording to both set of Eqs. {(1); (6)} and {(8); (9)} the graphicalillustrations of the non-collinear ferromagnetic structures called“umbrella” e and e0 are shown in Fig. 2 where the (π) planeassociated to the glide plane c.3 is chosen. For the sublattice C1, thespins S1 and S7 are assumed in the same side from the cubic axis[1 0 0]; for the sublattice C0

1, the spins S4 and S10 are assumed in thesame side from the cubic axis [0 1 1]. Furthermore, the projection ofthe rhombohedral direction [111] (g1) on the plane (1 1 1) isequivalent to the low symmetry axis [211] defined by the unit vectorp1. We precise finally the expressions of the unit vectors (qj) (j¼1–3)which are associated to the perpendicular axes {[011], [101], [110]}to [1 1 1], say outside the three glide planes but in the (1 1 1) plane

q1 ¼1ffiffiffi

2p ð� jþkÞ; q2 ¼

1ffiffiffi

2p ðþ i�kÞ; q3 ¼

1ffiffiffi

2p ð� iþ jÞ ð10Þ

Taking into account the general condition (faf 0 and aaa0), fourrhombohedral models around [1 1 1] axis of “double umbrella” typeare deduced and illustrated in the graphical schemes of Fig. 3a, b,c and d.

The best model of Fig. 3a used at 5 K by one of us [21] will beconsidered now with the specific condition for the parameters aand a0 (a¼a0) due to the absence above 13 K of the pure magneticlines (2 0 0)n and (600, 442)n. The set of Eqs. {(8); (9)} can bewritten as follows

Sj0 ¼ Sj0 þ6 ¼ ðf �a=3Þn�ð2ffiffiffi

2p

=3Þapj ð11Þ

Sj0 ¼ Sj0 þ6 ¼ ðf 0 þa=3Þnþð2ffiffiffi

2p

=3Þapj ð12Þwith j¼1–3 and j0 ¼4–6. For a better understanding of the specificcondition between the parameters a and a0 (a¼a0) applied on thespins C1{S1; S7} and C0

1{S4; S10}, it is more convenient to define therepresentative magnetic moments m1 and m0

1 by their “parallel //”(along [1 1 1] axis) and “perpendicular ?” (to [1 1 1] axis andoriented along [211] axis) components respectively. The compo-nents (m1//; m0

1//) and (m1?; m01?) are deduced using the

corresponding spherical coordinates (m1, θ1, ϕ1) and (m01, θ0

1,ϕ0

1) respectively. The angles (θ1; θ01) with respect to the unit

vector n of the [1 1 1] axis are the alone variable parameters in the(π) plane. The angles (ϕ1; ϕ0

1) relative to the unit vectors q1 in the(1 1 1) plane are fixed due specific condition obtained for TbIG(ϕ1¼1801 and ϕ0

1¼01) and the adapted model of Fig. 3a is thendefined with the following set of Eqs. {(13); (14)}

ðC1ÞS1 ¼ S7 ¼m1 ¼m1==nþm1?p1 ð13Þwith m1//¼(f - a/3)¼m1 cos(θ1) and m1?¼- (2

ffiffiffi

2p

/3)a¼m1 sin(θ1)and,

ðC01Þ S4 ¼ S10 ¼m0

1 ¼m01==nþm0

1?p1 ð14Þ

with m01//¼(f0 þ a/3)¼m0

1 cos(θ01) and m0

1?¼(2ffiffiffi

2p

/3)a¼m01 sin

(θ01).Taking into account the magnetic structure factors FM{c}(2 0 0)n

and FM{c}(600, 442)n associated to non observed pure magneticlines {2 0 0}n and {600, 442}n for T413 K, the expressions of theaverage calculated intensities are written as follows

IfcgM f200gncal ¼ 288η2f2ðTb3þ Þðm1? þm01? Þ2 ¼ 0 ð15Þ

IMcf g 600;442f gncal ¼ 897:88η2f2ðTb3þ Þðm1? þm0

1? Þ2 ¼ 0 ð16ÞOur experimental condition (a¼a0) is then defined finally by thefollowing equality for TbIG

m1? ¼ �m01? ð17Þ

The details of the refinements of crystallographic and magneticstructures by means of the Rietveld method of the FullProfprogram package [27] will be published elsewhere in order totake account the results of the refined NPD patterns collectedunder a low applied magnetic field and at various temperatureson another diffractometer at HANARO in KAERI, Korea, with awavelength λ¼1.8348 Å. For a more complete presentation of theresults in the 4.2–300 K temperature range, the refined para-meters by means of the Integrated Intensity method of the Fittingprogram [37] obtained from the NPD patterns recorded previouslyby one of us [24,25] with a wavelength λ¼2.49 Å have been added.It must be noted that the best fitted parameters give rise tocalculated magnetizations MS

cal along the three crystallo-graphic directions [1 1 1], [1 1 0] and [1 0 0] in agreement to thatobtained previously by magnetic measurements MS

mag [24,25,34,35].The thermal variations of the parallel (m1//; m0

1//) and perpen-dicular (m1?; m0

1?) components are reported in the (left) and(right) of Fig. 4 respectively. Below 160 K, the temperature depen-dences of the parallel m1// and m0

1// components present the sameabrupt increasing at 68 K with a maximum value of the reducedparameter, ΔmTb//¼ |m1// - m0

1//|¼1.57 μB/ionTb3þ at 54 K. Theopening of the “double umbrella” along [1 1 1] axis which is thedirection of the exchange interactions (Jcd) takes place near 160 Kwith decreasing temperature. The magnetic moments m1 and m0

1

remain in the glide plane c.3 up to 5 K according to the tempera-ture dependences of the components m1? and m0

1? .One expects more anomalous behaviors if the thermal varia-

tions of the cubic components of the spins C1 (m1x; m1y; m1z) andC0

1 (m01x; m0

1y; m01z) of the magnetic moments m1 and m0

1 areplotted as shown respectively in the (left) and (right) of Fig. 5.

Above 160 K, the following equalities (m1x¼m1y¼m1z andm0

1x¼m01y¼m0

1z) are in agreement with the Néel ferrimagneticstate existing along the easy magnetization axis [1 1 1] whereasa large local anisotropy appears below 160 K. The temperaturedependences of the m1x and m0

1x components along [1 0 0] axiswhich is the anisotropy direction of the crystalline field [28–30]have been found completely different. The thermal variation ofthe m1x component is similar to those of the m1// and m0

1//

components with the same order of magnitude in average and the

Fig. 2. Non-collinear ferromagnetic structures of the RE3þ ion spins: C1 in theWyckoff position (6e) called “umbrella” e; C0

1 in the Wyckoff position (6e0) called“umbrella” e0 .







same abrupt increase between 54 and 68 K during decreasingtemperature in contrast with the temperature behavior ofm0

1x whichis anomalous with a broad maximum occurring near 54 K. The m1y or

m1z components less steeply increase with decreasing temperaturethan that of the m0

1y or m01z components with an inflexion point

near 54 K.

Fig. 3. Four models of the “double umbrella”magnetic structures for the RE3þ moments of the irrep A2g. The model of Fig. 3a was used by one of us in the refined pattern at 5 K [21].

Fig. 4. Thermal variations of the parallel (m1//; m01//) (left) and perpendicular (m1?; m0

1?) (right) components obtained from the refined NPD patterns using the “doubleumbrella” model of Fig. 3a. The solid triangles point-up orange and point-down gray are the corresponding results at 109 and 127 K from the Néel model with the blackvertical lines as the error bars of order of 70.5–0.3 and 70.7–0.6 μB/ionTb3þ respectively [24,25]. The solid triangles point-up green and point-down blue indicatecomparable refinements with the solid triangles point-up wine and point-down red. The dash lines are guides for the eyes. (For interpretation of the references to color inthis figure legend, the reader is referred to the web version of this article.)

Fig. 5. Thermal variations of the cubic components of the spins C1 (m1x; m1y; m1z) (left) and C01 (m0

1x; m01y; m0

1z) (right) obtained from the refined NPD patterns using the“double umbrella”model of Fig. 3a. The solid triangles point-up orange and point-down gray are results from the Néel model at 109 and 127 K [24,25]. The solid spheres blueand green indicate comparable refinements with the solid spheres red and wine with the black vertical lines as the error bars of order 70.5 and 70.4 μB/ionTb3þ

respectively. The dash lines are guides for the eyes. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)







The temperature evolution of the calculated “double umbrella”obtained in Ref. [38] shows a divergence with our results: while thefirst umbrella closes, the second opens with increasing temperature.At the opposite, our results are in agreement to the anomalousbehaviors observed previously on TbIG single crystal without appliedmagnetic field in the acoustic properties by Kvashnina et al., [39] andelastic constants by Alberts et al., [40]. The longitudinal velocitiesof elastic waves behave anomalously in the temperature range130–140 K with minima for [1 1 0] and [1 1 1] crystallographic direc-tions while a maximum is found along the [1 0 0] direction [39]. Incontrast to the longitudinal velocities of the waves along [1 1 0] and[1 1 1], a minimum velocity is observed at�60 K along [1 0 0]. Thetemperature dependences of the longitudinal elastic constants behavealso anomalously in the temperature range 150–165 K with minimafor [1 1 0] and [1 1 1] crystallographic directions whereas along the[1 0 0] direction, a maximum is found [40]. In contrast to the long-itudinal elastic constants along [1 1 0] and [1 1 1], a minimum ofelastic constant is observed at�50 K along [1 0 0].

The abrupt increasing of all the components with the exceptionof m0

1x and m1y (or m1z ) is similar to that observed below�200 Kin the rhombohedral structural distortion from Ia3d along the easymagnetization axis [1 1 1] previously observed by XRD measure-ments [11,28–30,41,42]. In these works, the distorted cell wascharacterized by two parameters: angular distortion β¼901 - αR

equal to 70 30″ [11,28–30] and 100 76″ K [41,42] below 16 K and at21 K respectively, with αR, the axial angle of the rhombohedron;the cell edge distortion Δa¼ |ac - aR|¼28�10�4 Ǻ at 4.2 K [28–30],with aR, the rhombohedral lattice parameter. An anomalouscontraction of the cell side parameter Δa near 45 K [28–30] anda negative dilatation never seen before in REIG, except on TbIGsingle crystal near 80 K with an anomaly around 45 K [41,42] wereevidenced. The onset of the huge magnetostriction coefficientλ[1 1 1] which reaches the enormously large value of 2400�10-6

at 4.2 K [28–30,43] starts to develop at�150 K for powder sample[28–30] and 170 and 190 K for single crystal and heated singlecrystal respectively [42]. Below 160 K, the magnetostriction coeffi-cients λ[1 1 1] and λ[1 0 0] have the same positive sign, while λ[1 0 0]

passes through zero above 160 K [44]. A peak appears between 50and 70 K in the previous thermal variations of the followingrelation, λε,2(Tb3þ)exp/λ

ε,2(Tb3þ)cal, the quotient of the experi-mental values of the magnetostriction constant along [1 1 1] andthose derived theoretically from the one ion model [45]. Similaranomalous behaviors have been also reported in the case when alow magnetic field is applied (H¼5 and 0.1 kOe) as the kink-likeanomaly around 60 K observed in the recent magnetizationmeasurements [46]. From the temperature dependence of thesusceptibility of the Cotton-Mouton effect ΔφCM/ΔH [47] along[1 1 1] at H¼18 kOe, an anomalous maxima was observed near45 K with a rapid decreasing with increasing temperature upto�200 K.

There is a great similarity between evolution with T of the“double umbrella” and the lattice distortion. However, the “doubleumbrella” magnetic structure and the huge magnetostriction seemnot directly correlated. The occurrence of the magnetic reorderingof the Tb3þ magnetic moments at�160 K is apparently due to thelarge spin-orbit (LS) coupling where the rotation of the spins of thesublattices Cj and C0

j with (j¼1–3) which remain in the three glideplanes c, c.3 and c.32 up to 5 K should induce a rotation of theorbital moments. It seems that the onset of the increased of theanisotropy of both magnetization (To100 K [34]; To120 K [35])and magnetic susceptibility (To150 K) [5] is related to theappearance of the increased of the local anisotropy which becomesnoticeable below 160 K. This specific anisotropy temperature istentatively identified to the previous so-called “momentum angu-lar compensation point” (TJ) [48] predicted for three REIG (RE¼Tb,Dy and Ho) and localized for the Tb3þ at 150 and 190 K with the

assumption of the free and quenched ion values respectively. Itmust be noted that a similar correlation has been found recently inDyIG [49] between anisotropy of magnetizations and the occur-rence of the “double umbrella” magnetic structure.

The intricate behavior of the parameters of the “double umbrella”and the broad anomaly near 54 K are interpreted partially on thebasis of the theoretical and experimental works of Belov on ferri-magnets as REIG possessing a “weak-sublattice” of the RE3þ ions[50,51]. The observed maxima in the temperature dependences ofthe forced magnetostriction dλ/dH curves have been attributed tothe “paraprocess effects” at the so-called “low-temperature point”(TB). This point has been labeled for the first time by Pauthenet[9,10] as the “locking-temperature” (θb) from the temperaturedependences of the inverse of magnetic susceptibility χ-1 of mostheavy RE3þ ions (Gd, Tb, Dy, Ho and Er). At this critical point, a“sharp change of the long-range magnetic ordering in the “weak-sublattice” occurs where the interactions energy of the RE3þ ionswith the effective exchange field is comparable with the thermalenergy. The calculations in the molecular field approximation yieldthe following relation

TB �μBgSSðHexÞeff

kBð18Þ

where gS is the Landé-g factor of the spin S of the RE3þ ions and(Hex)eff, the effective exchange field induced by the “strong-sublattice” (mainly the (d) sublattice Fe3þ ions or what is the same,by the unidirectional exchange anisotropy of the Jcd interactions).The first evaluation of TB (7075 K) [9] is larger than the estimatedvalue (58 K) [50] which is in good agreement with our determina-tion (54 K). However, specific heat measurements Cp(T) performedpreviously on TbIG powder sample [52] yielded the experimentalvalue (45 K) deduced from the peak on the excess magneticcontribution Cp

ex(T) and the “low-temperature anomaly” coexistswith the anomalous “Schottky effect” due to the anisotropic crystal-line field. The magnetic non-cooperative Schottky anomaly is causedby redistribution of 4f-electrons of Tb3þ ions over excited levelsduring the increasing temperature. Taking into account the morerecent study of the thermodynamic properties of TbIG [53], compar-able temperature evolutions of the magnetic and Schottky contribu-tions to Cp(T) were observed in the 5–200 K range with twosuperposed and confounded anomalies near 50 K. We conclude thatthe anomalous temperature behavior of the “double umbrella”below 160 K can be explained adequacy with the large LS couplingwhich disturbs the unidirectional exchange anisotropy induced bythe “strong ad sublattice” occupied by Fe3þ ions on the “weak e, e'sublattices” occupied by Tb3þ ions. The partial but sharp change ofthe long-range magnetic ordering which occurs in both sublatticesof the Tb3þ ions at �54 K in agreement with the predicted valueTB¼58 K could be also explained by the competition at lowtemperatures between the thermal motion and the strong parapro-cess which has been termed “exchange-enhanced paramagnetism”

[50] with other concomitant anisotropy effects having a magneticnon-cooperative origin (crystalline field interactions).

Hur et al., [5] have suggested a possible scenario for the low-field MD effect in TbIG in which the field-induced reorientation ofthe rhombohedral domains gives rise to the change of themacroscopic electronic and/or ionic polarizability, which deter-mines the parameter ε. We believe that the low-field MD effectwhich appears in TbIG below 150 K and the coupling betweenexchange magnons and ligand-field excitations of the Tb3þ ions inthe 60–80 K temperature range are related to the anomalousbehavior of the “double umbrella” near TJ�160 K and TB�54 K.They could be caused not only from a sample change dimensiondue to magnetostriction below TJ but also from an important spin/lattice coupling near TB.







4. Conclusion

The magnetic reordering in TbIG has been investigated belowroom temperature using symmetry considerations of Bertaut inthe interpretation of the observed NPD patterns. The magneticstructure belonging to the Néel model along the [1 1 1] axisevolves below 160 K toward a “double umbrella” model up to13 K. An abrupt change in the long-range magnetic ordering of theterbium moments belonging to the “weak-sublattice” occurs at the“low-temperature anomaly” TB�54 K. The method of the “sym-metry lowering device” of Bertaut shows that both ferromagneticmodes, collinear and non-collinear could be described within theirrep A2g of the highest subgroup R3c derived from the singlethree-dimensional irrep Г4g¼T1g of the space group Ia3d: thecorresponding rhombohedral magnetic symmetry is R3c'in the5–300 K temperature range.

The occurrence of the ME and MD effects observed in thiscompound near 150 K could be related with the manifestation ofthe anisotropic magnetic character of the “double umbrella” whichis noticeable below 160 K labeled as the TJ point. The couplingbetween exchange magnons and ligand-field excitations of theTb3þ ions observed between 60 and 80 K in the study of the farinfrared transmission properties in TbIG could be associated to theanomalous behavior of the “double umbrella” in the region of thelow-temperature anomaly at the TB point. The NPD patterns undera low external applied magnetic field associated to Cp(T) measure-ments performed on single crystal will be able to determine thenature of the changes around TB�54 K and to ascertain thesuggested scenario attributed to low-field MD effect in TbIGknowing that in this situation, the magnetic domains, which arestrongly coupled to the lattice distortions, can not be readilyreoriented by H of only 1–2 kOe.

Acknowledgements

The present paper is dedicated in honour of the late of theAcademician Pr. Dr. Erwin Felix Lewy Bertaut (1913-2003) and inmemory of his scientific achievements and cornerstones on grouptheoretical techniques in magnetic structures analysis.

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Anomalous magnetic reordering in magnetodielectric terbium iron garnet at low temperatures