ON THE MAGNETIC BEHAVIOUR OF HEAVY RARE-EARTHS RCo … · energies balance, while the magnetic...

ON THE MAGNETIC BEHAVIOUR OF HEAVY RARE-EARTHS RCo2

COMPOUNDS

E. BURZO1,2, L. CHIONCEL3,4

1Romanian Academy, Cluj-Napoca Branch, 400015, Cluj-Napoca, Romania

E-mail: [email protected] 2Faculty of Physics, Babes-Bolyai University, 400084 Cluj-Napoca, Romania

3Augsburg Center for Innovative Technologies (ACIT), D-86135 Augsburg, Germany,

E-mail: [email protected] 4Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute of Physics,

University of Augsburg, D-86135 Augsburg, Germany

Received December 4, 2017

Abstract. The magnetic behaviour of RCo2 (R = Tm, Er, Ho, Tb) compounds,

above and below the Curie temperatures, Tc, is analysed. In the magnetic ordered

phase, the exchange interactions (with spatial extension within the unit cell) are

discussed in connection with R5d band polarizations. It is also shown that the

exchange interactions, at T > Tc, are not sufficiently strong to induce the short-range

magnetic order (of Griffiths-type) inferred from SANS experiments. The successive

changes in the orientations of R and Co moments, as function of temperature and

external field can be well described by a phenomenological model based on the

energies balance, while the magnetic behaviour of cobalt is at best described by the

spin fluctuations

Key words: rare-earth compounds, magnetic properties, exchange interactions.

1. INTRODUCTION

The RCo2 compounds, where R is a rare-earth or yttrium crystallize, at

ambient conditions, in a cubic-type structure having Fd m space group. Below the

Curie temperatures, Tc, their crystal structures are distorted due to magnetostrictive

effects [1].

The RCo2 compounds, when R is a magnetic heavy rare-earth (R = Gd to

Tm) are ferrimagnetically ordered, the rare-earth and cobalt moments being

antiparallelly oriented. The magnetic transitions are of first order when R = Dy,

Ho, Er and of second order for R = Gd, Tb, Tm. Above the Curie points, the

reciprocal susceptibilities, χ-1

, follow non-linear temperature dependences as

expected for ferrimagnetic systems [2, 3]. The paramagnetic data were analysed

considering that cobalt has an exchange enhanced magnetic susceptibility [4] or an

effective intrinsic moment [2, 3].

Romanian Journal of Physics 63, 601 (2018)

mailto:[email protected]

mailto:[email protected]

Article no. 601 E. Burzo, L. Chioncel 2

The polarized neutron diffraction studies performed on RCo2 compounds

with R = Tb [5], Ho [6, 7] and Tm [7, 8], above the Curie points, Tc, in the

presence of external fields, evidenced an antiparallel orientation of rare-earths and

cobalt moments. A change in the direction of cobalt moment from antiparallelly to

parallelly oriented to Tm one was shown in TmCo2, as temperature increased

[7, 8]. The above data were analysed assuming that cobalt has an exchange

enhanced magnetic susceptibility, the cobalt moment being induced by the total

field acting on the cobalt atoms [5–8]. The exchange field acting on cobalt, at

T > Tc, is smaller than the value characteristic for metamagnetic transitions,

Hc 75 T [9–13]. Thus, only an intrinsic effective cobalt moment can be present.

This statement has been confirmed particularly by analysing the thermal

dependences of magnetic susceptibilities for RCo2 compounds with non-magnetic

[14, 15] as well as magnetic [16, 17] rare-earths. The following studies, performed

on ErCo2 compound evidenced, in the presence of external field, the same

antiparallel orientation of Er and Co moments, at T > Tc [18–20]. An intrinsic

cobalt moment was also shown [18]. In addition, the SANS measurements were

interpreted as evidence for short-range correlations, with a length of 7 Å [21, 22].

These were associated with the presence of Griffiths-like phase above the Curie

point. The same antiparallel orientation of rare-earth and cobalt moments, at T >

Tc, in the presence of external field, was further reported in RCo2 (R = Ho, Tm)

compounds [23–25], in agreement with previous studies [6–8]. The Griffiths phase

was described as “establishing a short-range order a kind of low temperature

remnant magnetic order of undiluted system” [24]. The magnetic behaviour

characterized by an antiparallel orientation of rare-earth and cobalt moments, in the

presence of external field, at T > Tc, was called parimagnetism [24]. As function of

temperature and external field, different arrangements of cobalt and rare-earths

moments, above Tc, were also shown [23–25].

To correctly capture the electronic structure of strong magnets such as the

rare-earth transition-metal RCo2 compounds, the Density Functional Theory (DFT)

[26–28] needs further development. The discrepancy between the measured and

computed quantities such as local moments, exchange interactions or

magnetocrystalline anisotropy arises from the strong intra- and inter-atomic

correlations induced by Coulomb interactions that are not sufficiently well captured

by the Local Density Approximation [27, 29, 30] or its variants. In the recent years,

Dynamical Mean field theory (DMFT) [31–33] and its extension the LDA+DMFT

method [34, 35] is able to include such subtle effects of correlations. This method

has been successfully applied to the calculation of neutron magnetic form factors of

actinides by applying an external magnetic field [36], as well as the bulk and

surface quasiparticle spectra and the orbital magnetism [37] in Fe, Co, and Ni

metals, or YCo2 [38] to name just a few. Technical difficulties (Hilbert space size,

3 On the magnetic behaviour of heavy rare-earths RCo2 compounds Article no. 601

accurate numerical implementation) hinder the current methodological

development to be applied to the RCo2 compounds. Therefore, phenomenological

theories that explain the physical properties in these systems taking into account

the exchange interactions and paramagnetic susceptibilities, magnetic ordering

temperature and magnetization, in a unitary way, is still valuable. In the present

paper, starting from the analysis of exchange interactions as well as of cobalt

magnetic behaviour, at T > Tc, we show that in RCo2 (R = Tb, Ho, Er, Tm)

compounds no short-range magnetic order can exist as previously reported

[18–25]. Therefore, no magnetic ordered phase, of the Griffiths-type, can be

shown, at T > Tc, the short range correlations observed by SANS being associated

with the exchange coupled magnetic atoms inside a region with dimension of

7–8 Å the same as the RCo2 lattice constants (7.2–7.4 Å). The orientations of

rare-earth and cobalt moments, above Tc, in the presence of external field, can be

well described taking into account the balance of thermal and magnetic energies, at

finite temperature.

2. EXCHANGE INTERACTIONS IN RCo2 COMPOUNDS

The exchange interactions in RCo2 compounds, at T < Tc, are rather

complex, essentially determining the cobalt moments. Those between R and Co

atoms are described as of 4f–5d–3d type [39, 40]. The exchange interactions

between cobalt atoms extend to nearest neighbours. Since of large extension of

R5d orbitals, direct interactions are expected to take place also between R atoms

[41], in addition to those by means of conduction electrons.

Fig. 1 – The R5d band spin-polarizations ( – LDA, – LDA+U) in heavy rare-earths RCo2

compounds. The inset represents the schematic path of exchange interactions within the unit cell.

Solid/dashed black arrows represent the direct/reverse path of the 5d-3d interaction. The results

of the R5d spin-polarization were taken from the reference [40], where the details

of the LDA/LDA+U calculations are also discussed.


The band structure calculations were performed on RM2 (M = Fe, Co, Ni)

compounds with heavy rare-earths [40]. The R5d band spin polarizations, M5d, in

RCo2 compounds, as function of De Gennes factor, G, are plotted in Fig. 1. These

are antiparallelly oriented to cobalt moments and can be described by two additive

contributions:

M5d = M5d(f) + M5d(d). (1)

The first one, M5d (f) = αG, is due to local 4f–5d exchange, while the second

one, M5d(d), is the result of R5d-Co3d short range exchange interactions, the

corresponding hybridizations effects, respectively. The M5d(d) contributions to R5d

band polarizations were shown to be proportional to i i iz M [40]. By zi is denoted

the number of cobalt atoms situated in the first coordination shell to an R one and

Mi are their magnetic moments. This relation is followed also in pseudobinary

RCo2 alloys as evidenced in Fig. 2. The slope of this linear dependence, |M5d(d)|/

i i iz M 2·10-2

, is the same as that evidenced in RM2 (M = Fe, Ni) – based

compounds, having similar crystal structures as RCo2 ones. These data show a

significant interdependence between the R5d band polarizations and cobalt

magnetic moments. As already mentioned, the part played by R5d band

polarization in analysing the interactions in RCo2 compounds being a fundamental

question [40]. The Curie temperatures of RCo2 compounds follow a linear trend as

function of M5d values.

Fig. 2 – The M5d(d) components of R5d band polarizations, in RCo2-pseudobinary compounds,

as function of i i iz M .

The above data suggest also that the exchange interactions are

interdependent, within the spatial extension of the unit cell. The induced cobalt

moment, by 4f–5d–3d exchange path, is stabilized by the Co3d-Co3d direct

exchange interactions. Concomitantly with the appearance of a cobalt moment, an

additional polarization, M5d(d), is induced on R5d band by a reverse path, as


depicted in the inset of Fig. 1. The cobalt moments, parallelly with R5d band spin-

polarizations, are strongly influenced by magnetic dilution effects, both at R and

Co sites, as evidenced for example, in Gd(Co1-xNix)2 [42, 43] or (GdxY1-x)Co2 [44]

pseudo-binary compounds. The above complex and interdependent magnetic

interactions, which starts from an “atomic-like” picture, can be extended into real

space up to the unit cell dimensions, and one may be tended to associate this

behaviour to a “cluster” of magnetic interacting atoms. Within the proposed short-

range order phenomenological picture [21, 22], above the Curie temperatures, the

thermal energy might not be high enough to compensate the exchange energy and

thus some remnant magnetic coupling between atoms would exist, which was

proposed as interpretation for the SANS experiments [21, 22]. In general, the short-

range magnetic order can be important in the description of thermodynamic

properties, in the vicinity of magnetic phase transitions. It influences also the

structural phase transitions, since the presence of short-range magnetic order in one

of the phases, will affect conditions of appearance of a new (another) phase with

changing temperature, pressure or impurities concentration. In the context of the

RCo2 compounds, in previous studies, no structural change or any other

thermodynamic transformation, nor magnetic frustration phenomena, or

dimensionality reduction have been reported, at T > Tc, which might indicate that

the physical conditions for the appearance of a short-range order are not realized.

Thus, for the RCo2 compounds the short-range order model, might not be suitable,

or at least is incomplete, from our point of view. From a theoretical perspective, a

clear knowledge of the exchange interactions is required to interpret the

experimental data. The main difficulty is the precise theoretical expression for the

exchange interactions in the specific materials. Still, one of the best approach is

based on the local environment model [42, 43]. However, as will be shown, this

also does not support the short-range model for the materials in discussion.

3. MAGNETIC BEHAVIOUR OF RCo2 COMPOUNDS

WHERE R = Lu OR Y

The magnetic properties of RCo2 compounds, above the Curie points, are

determined also by those of cobalt. As mentioned in introduction both the presence

of an exchange enhanced paramagnetism [4] or an intrinsic effective moment [2, 3]

were initially proposed. Latter studies, evidenced that the cobalt magnetic

behaviour, in RCo2 compounds, can be described by the spin fluctuations model

[14–17]. The model [45, 46], takes into account the balance between the

frequencies of longitudinal spin fluctuations, which are determined by their

lifetime and of transverse fluctuations that are of thermal origin, leading to the

concept of temperature induced moment. For a nearly or weak ferromagnet, as

cobalt in RCo2 (R = Lu, Y) compounds, the wave number dependent susceptibility,

χq, has a large enhancement due to electron-electron interactions for small q values.


The average amplitude of spin fluctuations 2

loc B qS k T χ increases with

temperature and reaches an upper limit, at a temperature T*, determined by the

charge neutrality condition. At T > T*, a Curie-Weiss behaviour is predicted similar

as in systems having local moments. The moments are localized in q-space

[45, 46].

Fig. 3 – The temperature dependence of the mean square root amplitudes of the fluctuating cobalt

moments, in RCo2 (R = Lu, Y) compounds. By broken lines and points are plotted the experimentally

determined effective cobalt moments in the corresponding temperature ranges. The effective cobalt

moments determined from already published data, at lower temperatures, on ErCo2 [18], HoCo2 [25]

and TmCo2 [25] are also plotted.

Starting from the calculated density of states and taking into account the

effect of spin fluctuations, the temperature dependences of 2

locS were calculated

in YCo2 [47] and LuCo2 [48] compounds, by using the procedure previously

reported [49, 50]. As can be seen in Fig. 3, there is a tendency to saturate 2

locS

values, at temperatures T > 550 K, where a linear χ-1

vs T dependence has been

experimentally observed. The same behaviour was also reported using results from

the band structure calculations in YCo2 compound [51].

4. MAGNETIC BEHAVIOUR OF RCo2 COMPOUNDS

WITH MAGNETIC HEAVY RARE-EARTHS

4.1. THE NON EXISTENCE OF A GRIFFITHS-TYPE PHASE AT T > Tc

Magnetic measurements were made on ErCo2 compound at temperatures

where the Griffiths phase is supposed to be present [2, 17]. As seen in Fig. 4, linear

dependences of the magnetic moments on the external field are shown, evidencing


that no magnetic ordered phase is present at T > Tc. The Curie temperature of

ErCo2 is shifted to higher values as the external field increases [52]; for instance in

a field of 70 kOe, a value Tc = 42 K was obtained, while in a smaller external field

(H < 0.5 kOe) a value Tc = 33 K was determined. As a result, at temperatures

T > Tc = 33 K, deviations from the linear field dependence of magnetization are

possible in high magnetic fields. Magnetic ordered impurities, if exist, are below

0.1% and influence very little the experimental data [2, 3, 17]. The same behaviour

can be shown in HoCo2, where both Ho and Co magnetic moments are linearly

dependent on the external field, as evidenced by neutron diffraction studies [6, 7] –

Fig. 5.

Fig. 4 – The field dependence of magnetizations in ErCo2, at some temperatures above Tc.

Fig. 5 – The field dependence of cobalt and holmium moments in HoCo2

as determined by neutron diffraction study. The data are taken from reference [6].

These data suggest that there are no magnetic ordered cobalt clusters, of a

Griffiths phase type at T > Tc. The above statement is also confirmed by the

evolution with temperature of the rare-earths and cobalt moments, at T > Tc, as


determined by polarized neutron diffraction, in various external fields [5–8] –

Fig. 6. There is a linear dependence of cobalt moments on the rare-earth ones in the

studied temperature range (40 K < T < 300 K) and external fields (10 kOe < H

< 57.2 kOe). The slopes, a = |MCo/MR|, are dependent on the rare-earth partner,

decreasing in the same way as the M5d band polarizations, as evidenced in Fig. 7.

On the same figure, the a 0.041 value, as obtained by the polarized neutron study

on ErCo2 single crystal, at T = 40 K and H = 50 kOe, is given [24]. The a values

obtained from reported data by Givord et al. [5–8] result from studies performed in

an extended ranges of fields and temperatures while that obtained by [24] refers to

only one value. This can explain the little difference from the expected trend,

denoted by solid line in Fig. 6.

Fig. 6 – The cobalt magnetic moments in RCo2 (R = Tb, Ho, Tm) compounds as function

of rare-earths ones as determined from polarized neutron diffraction.

The data used are from references [5–8].

Fig. 7 – The relation between a = |MCo /MR| parameters and the M5d band polarizations.


The 4f–5d–3d intra-atomic exchange interactions may still be present, in a

relative large temperature range, above the Curie temperature, Tc. As already

mentioned [40, 53], the R5d-Co3d short range exchange interactions are important

in describing the magnetic properties of rare-earth transition metal compounds,

which is the essential idea of the Campbell model [39]. These R5d- Co3d couplings

may still exist above Tc, but their intensities are not so high to induce a magnetic

ordering. The presence of La5d-Co3d magnetic coupling was also shown in

paramagnetic LuCo2 single crystal, at T = 100 K and field of H = 57.2 kOe, by a

polarized neutron diffraction study [54]. A cobalt moment of 0.016 B has been

determined, the corresponding form factor being similar to that of 3d electrons in

cobalt metal. The Lu5d band is negatively polarized and of 0.007(5) B, the ratio

|M5d/ i i iz M | (3.5 2.5)·10-2

being, in the limit of experimental errors, the same

as that determined in magnetic ordered RCo2 compounds, in the ground state

(2·10-2

). The above data suggest that the induced cobalt moments by external fields

(H < 57 kOe), at T > Tc, are rather small. This statement is also confirmed by the

linear field dependences of cobalt moments in the paramagnetic range as

evidenced, for example, in HoCo2 – Fig. 5. As a consequence, no such effect will

be considered.

In the followings, we explore the possibility that the Griffiths phase is

connected with the existence of a quantum critical point. In the last decades in

condensed matter physics, a new class of phase transitions, called quantum phase

transitions, taking place at absolute zero has been predicted for materials

containing rare-earth elements or Kondo systems. A quantum critical point is a

point in the phase diagram of a material where a continuous phase transition takes

place at absolute zero. A quantum critical point is typically achieved by a

continuous suppression of a nonzero temperature phase transition to zero

temperature by the application of a pressure, field, or through doping.

Conventional phase transitions occur at nonzero temperature when random

thermal fluctuations leads to a change in the physical state of a system. In the

absence of the thermal fluctuations which trigger conventional phase transitions,

quantum phase transitions are driven by the zero point quantum fluctuations. As

the pressure controls the quantum phase transition (QPT), in rare-earth

intermetallics, a quantum critical point (QCP), separating a paramagnetic phase

from a magnetically ordered one, occurs [55]. The Grüneisen ratio [56],

(T)(T)/C(T), where (T) and C(T) denote the contributions of volume thermal

expansion and electronic specific heat respectively, diverges at the approach to any

pressure sensitive QCP [55, 57]. In the spin wave theory, the divergence is given

by cr 1/Tε, where ε = 1/z, the being the spatial correlation length exponent

and z is the dynamical exponent.


Fig. 8 – The volume thermal expansion coefficient, β/T, and the specific heat C/T as function

of temperature for ErCo2 compound. The solid lines represent the fit according to the 3D-SDW

scenario. The data were obtained from the temperature dependences of lattice parameters [58]

and of specific heat o [59], ★ [60].

In the search for the possibility of the Griffiths phase, we combine the

experimental results of specific heat and thermal expansion and look for the

correlation length exponents for some of the RCo2 compounds. The neutron

diffraction studies on ErCo2 [17] and HoCo2 [16] evidenced a high sensibility to

pressure of the magnetic transition temperature of cobalt sublattice magnetization.

As a consequence, the evolution with temperature of the Γ values, in these

compounds, can gives information on the presence of any quantum critical point

connected with rare-earth and cobalt sublattice magnetic ordering. According to

[57], at any pressure sensitive QPC, the volume thermal expansion coefficient is

more singular than the specific heat since in the latter case there is a high

temperature tail of a (nuclear) Schottky anomaly when approaching to Tc on a

magnetic ordered state.

The ErCo2 thermal expansion [58] and specific heat [59, 60] were already

reported. The temperature dependence of the volume thermal expansion coefficient

is well described by the relation predicted by the three-dimensional Gaussian

scenario [55, 57], i.e. a sum of a (1/ T ) term and a constant contribution – Fig. 8.

In the temperature range 42 K < T < 300 K the experimental data follow the

relation (T)/T = a / T +b with a = 7.414·10-8

K-1.5

and b = –3.262·10-9

K-2

. The

temperature dependence of the electronic contribution to the specific heat could be

analysed also in the 3D-Gaussian QPC model. In the 53 K to 300 K temperature

range, the data were fitted as C(T)/T = c – d T with c = 0.906 J/molK2 and

d = –0.0376 J/molK1.5

– Fig. 8. Since T1/2

and C T3/2

respectively, it results

1/T. The Grüneisen exponent, ε = 1, describes the magnetic properties of RCo2

compounds in agreement with the 3D-SDW QPC prediction, as experimentally

confirmed by neutron diffraction studies [16, 17]. The divergence of Γ(T), at low


temperatures, close to Tc, suggests that only one magnetic ordered phase is present,

corresponding to the Curie point of ErCo2. To conclude, no quantum critical point

is realized in ErCo2 compounds at T > Tc.

4.2. MAGNETIC CONFIGURATIONS IN RCo2 (R = Tm, Er, Ho), AT T > Tc,

IN THE PRESENCE OF EXTERNAL FIELD. PHENOMENOLOGICAL MODEL

The magnetic measurements performed on RCo2 (R = Tm, Er, Ho) com-pounds, in a limited temperature interval above Tc, evidenced different orientations of rare-earths and cobalt moments as function of temperature and field intensity. An antiparallel alignment of R and Co moments, in the presence of external field, has been evidenced at Tc < T < T1 by polarized neutrons diffraction [5–8, 24], XMCD [18–20, 24, 25, 61] or determined from longitudinal and transverse susceptibilities [19, 20, 24, 62]. An illustration of the reorientation transition at T1 is given in Fig. 9.

The temperatures T1 depend on the rare-earth partner and external field

intensity. When no external field is present, the values T1 75 K (R = Tm), 100 K (R = Er) and 125 K (R = Ho) were reported [25]. Generally, the T1 temperatures decrease when increasing the external field. At T > T1, an interesting feature was found in RCo2 (R = Tm, Ho) compounds. The R magnetic moment is opposite to the cobalt one and also to the external field of low intensities. In high magnetic fields, a parallel orientation of the R and Co moments to the external field has been observed, as in classical ferrimagnets, at T > Tc – Fig. 9. The corresponding magnetic phase diagrams for TmCo2 and HoCo2 compounds are plotted in Fig. 10 by solid lines [25]. As a general trend, the external field necessary for imposing a parallel orientations of R and Co moments increases with temperature.

Fig. 9 – (Color online) Blue/red cones represent thermal fluctuating local moments

of rare-earth/cobalt in the paramagnetic state (T > Tc). At finite temperatures, T > T1, and increasing

applied external fields (black arrow) the R and Co moments projected along the field turns

from an anti-parallel orientation into a parallel one.


Fig. 10 – The temperature-field diagrams characteristic for observed magnetic arrangements.

The solid lines are guide for eye separating the different configurations of R (R = Tm, Ho) and Co

magnetic moments [25]. By points are denoted the calculated data using the magnetizations

determined by neutron diffraction and magnetic measurements, assuming anisotropy energies

Ea = 0.07 meV (TmCo2) and 0.09 meV (HoCo2).

The successive changes in the orientations of R and Co moments, at T > Tc,

in the presence of external field, can be analysed starting from thermal evolution of

the total energy, Et, characterizing the RCo2 compounds. In a first approximation,

this include the exchange energy, Eexch at the level of unit cell, Eth, the thermal

energies, Ea, the anisotropy energy and EH the energy involved by the interaction of

magnetic moments with external fields:

Et = Eexch + Eth + Ea + EH. (2)

In the absence of external field, at temperatures T1, the energy associated

with the interdependent exchange interactions at the level of unit cell, Eexch and the

anisotropy energy are compensated by the thermal one, kBT1. As a result, assuming

that the anisotropy energy, in paramagnetic phase is low, when Eexch kBT1, values

Eexch = 6.5 meV (R = Tm), 8.6 meV (R = Er) and 10 meV (R = Ho) were

estimated. These are significantly lower than the exchange interaction energy

between cobalt atoms, of 40 meV, as evidenced at T < Tc, in magnetic ordered

HoCo2 compound [63]. Exchange interaction energies between cobalt atoms in

RCo2 compounds, having more than one order of magnitude than the above, were

also estimated [25].

The external field acts to align parallelly the R and Co moments, opposite to

that imposed by the interdependent exchange interactions, extending within the

unit cell. As a result, the T1 temperatures will decrease in the presence of external

field. Starting from the magnetic moments, determined by neutron diffraction and

magnetic measurements, M, obtained in various fields, H, the changes in the re-

orientation temperature, ΔT1, in the presence of external fields, can be estimated

according to the relation kBΔT1 = 0MH. Values of ΔT1 between 2 K and 12 K


were obtained, as function of external field intensities (10 up to 50 kOe). This trend

is in good agreement with experimental evidence [25]. The changes in the orientation of R and Co magnetic moments, as function

of external field intensity, at T > T1, can be reasonable described by considering the presence of the anisotropy energy, Ea, mainly connected with rare-earth ions. The cobalt orbital contributions in RCo2 compounds, as determined by

59Co NMR, is of

the order of 5–9% [64], the involved anisotropy being rather small. The single ion anisotropy of the rare-earths is considerable greater, imposing their orientation along the easy axis of magnetization. While the cobalt moment easily orients parallel to external field, the rare-earth ones remain aligned to the easy axis when the energy of the interaction with magnetic field, EH, is smaller than the anisotropy one. The anisotropy energy, Ea, of the R ions, at T > Tc, can be estimated from the field necessary to align parallelly the Co and R moments – Fig. 10. Assuming that the exchange interactions, in limited temperature range above T1, are little dependent on temperature, the anisotropy energy, Ea, involving mainly R atoms can

be evaluated starting from the relation Ea = 0HaMi (i = Co, R). Since at T > Tc,

Mi = χiHa, we have a a 0 Co RH E /μ (χ χ ) . By using χCo and χR values determined

from neutron diffraction studies [8] or as obtained by magnetic measurements [2, 8, 16, 65], the Ha fields were estimated. A good agreement with the experimental data on TmCo2 is obtained, when Ea = 0.07eV. In case of HoCo2 compound, the phase diagram, at T > 175 K, can be well described assuming Ea = 0.09 eV, when using both neutron diffraction data [6, 7] or magnetic measurements [16]. The deviation from the expected behaviour in the temperature range 120 K < T < 175 K, can be correlated with the partial superposition with the interval where the moments are

antiparallelly oriented (Tc< T < T1 135 K) and also with the fluctuations of magnetic moments in the transition temperature range between the two possible magnetic configurations (T < T1, T > T1), when these can coexist. When increasing temperature, the projections of the magnetic moments along the field direction decreases. Thus, for compensating the anisotropy energy, the external field must to increase, as experimentally observed.

4.3. THERMAL VARIATIONS OF RECIPROCAL SUSCEPTIBILITIES

IN MAGNETIC ORDERED RCo2 COMPOUNDS

The thermal variations of reciprocal susceptibilities, χ-1

, of RCo2 (R = Tm,

Er, Ho, Tb) compounds are given in Fig. 11. In addition to already published data

on TmCo2 [65], ErCo2 [17], and HoCo2 [16], the χ values obtained from neutron

diffraction data [5, 7, 8] are also plotted. A good agreement between the two sets of

values can be seen.

The reciprocal magnetic susceptibilities, χ-1, follow non-linear temperature

dependences, as generally characteristic for ferrimagnetic compounds. A linear

behaviour is evidenced in the asymptotic region, T > 550 K. The determined Curie


constants, in the high temperature range, are greater than those of R3+

free ions,

suggesting contributions also from cobalt atoms. According to addition law of

magnetic susceptibilities and assuming that the effective rare-earth moments are

given by the free ion values, the contributions of cobalt atoms to the Curie

constants were determined [14]. The effective cobalt moments, Meff (Co) for

R = Y, Lu and Tm are close to Co2+

free ion value and decrease little when Curie

temperatures of RCo2 compounds, exchange field acting on cobalt atoms,

respectively increase – Fig. 3. This behaviour has been attributed to partial

quenching of spin fluctuations by internal fields [14].

Fig. 11 – Thermal variations of reciprocal susceptibilities χ-1 in RCo2 compounds

(R = Tm, Er, Ho, Tb). The χ-1 values obtained from neutron diffraction data [5–8] are also plotted.

The ratio r = Sp/So between the number of cobalt spins determined from

effective cobalt moments, Sp, and the saturation ones, So, follows a

dependence, as predicted by spin fluctuations model [66, 67] – Fig. 12. The r

values obtained in ErxY1-xCo2 compounds follow the same trend [68].

In some papers, the temperature dependences of magnetic susceptibilities

were analysed only in the temperature range Tc < T < 300 K, where χ-1

vs T plots

can be approximated by linear variations. The effective moments determined in this

temperature range are smaller even than those of free R3+

ions [21, 25]. At least up

to T1 temperatures, these magnetic susceptibilities, describe the antiparallel

orientation of rare earth and cobalt moments and as a result χ = χR–χCo. Thus, at

lower temperatures than T 500 K no reliable information can be obtained, from

the χ-1

vs T dependences, as in common ferrimagnetic systems, where at T > Tc,

χ = χR + χCo.

The effective cobalt moments, experimentally determined in RCo2

(R = Ho, Er, Tm, Lu) compounds, at T >550 K, are plotted by dashed lines in

Fig. 3. The computed effective moments in RCo2 (R = Y, Lu) compounds, in the


above temperature range, are somewhat smaller than the experimental ones [14].

Their saturation has been not fully attained since these were computed by using

only the first S2 term [47, 48] in series expansion of the magnetic susceptibilities

[49, 50].

Fig. 12 – The ratio r = Sp/S0 as function of –2/3

cT for RCo2 (R = Gd, Tb, Dy, Ho) and ErxY1-xCo2

compounds. The Sp values determined from Curie constants and So from saturation cobalt moment

(MCo 1 B).

The cobalt moments, at temperatures somewhat higher, but close to the

Curie points have been determined by magnetic circular dichroism. Values

MCo = 0.2 B (ErCo2) [18], 0.3 B (HoCo2) and 0.4 B (TmCo2) [25] were reported.

The cobalt moment determined in ErCo2 was shown to increase with temperature

[18], although, in a latter report, has been considered to be rather constant in some

temperature intervals [25]. The polarized neutron diffraction studies at T > Tc [5–8]

evidenced a decrease of the cobalt moments with temperature, these “moments”

really being the projections along the field direction. The above contradiction can

be reconciled as follows. There is an increase of the cobalt moments with

temperature at T > Tc, as expected in a spin fluctuations model, concomitant with

an apparent decrease due to thermal disordering, their projections on the external

field direction decreasing with temperatures. Thus, in some limited temperature

range, the Meff(Co) values can be really seen as relative constant. Since MCo = gS,

the spin values, S, corresponding to the already determined moments by circular

dichroism [18, 25] are S = 0.1, 0.15 and 0.2, respectively. The corresponding

effective moments Meff(Co) = g S(S+1) are of 0.66, 0.83 and 1.00 B, respectively.

These data are located, on Fig. 3, at the temperatures where have been determined.

As a whole, these also support the predictions of spin fluctuations model.

Finally, we note that around the temperature T1, where the R and Co moment

changes from antiparallel to parallel orientations, the magnetic susceptibilities of

RCo2 (R = Tm, Er, Ho) compounds, in small external field, deviate from smooth


dependences [18, 25]. The above trend can be correlated with fluctuations,

particularly of cobalt moments, when the internal field is rather small, the exchange

and thermal energies being nearly compensated. The normal behaviour is

recovered in somewhat higher external fields. These anomalies in the temperature

dependences of the magnetic susceptibilities cannot be seen if data are obtained at

fields higher than 0.5 kOe, as commonly used in determining magnetic

susceptibilities at T > Tc.

5. DISCUSSIONS

The present analysis suggests that the “ferrimagnetic correlations”, at

T > Tc, as evidenced by SANS experiments, having the correlation length of 7Å

are of the same real space extension as the unit cell dimensions of the RCo2

compounds. The magnetic interactions between constituting atoms within this

spatial extension may survive in the paramagnetic phase. However, the intensities

of these interactions are not enough to induce a short-range magnetic ordered state.

The temperature dependence of the Grüneisen ratio evidences only the Curie

temperature, and no quantum critical point, at T > Tc can be found in ErCo2. As we

detailed in the present paper a large amount of experimental data points towards

the validity of the spin-fluctuations model for the behaviour of Co in the

paramagnetic regime of RCo2 compounds. It is also true that spin fluctuations are

at the origin of random magnetic interactions above Tc. Although a heuristic

interpretation of the SANS data was proposed in favour of the applicability of the

Griffiths model, we believe, that the arguments presented are not strong enough for

the following reasons: (i) The antiparallel coupling of R and Co moments, at T >

Tc, can be evidenced mainly in the presence of external field, up to a temperature

T1, which depends on the strength of 4f–5d–3d magnetic coupling; (ii) The thermal

energy is not sufficient to induce full disorder, in the considered temperature range,

as in normal ferrimagnets. Parallel orientations of magnetic moments in external

fields can be seen only at higher temperatures; (iii) The evolutions with

temperature of cobalt moment and considering the anisotropy energy, particularly

of R ions, can determine a more complex magnetic arrangement of rare-earth and

cobalt moments in the presence of external field.

Analysing the magnetic properties of the clusters, as evidenced by SANS

measurements in ErCo2 compound, an effective moment Meff(cluster) 20 B has

been reported [19]. The “cluster” was considered to be constituted from 60 to 100

cobalt and 30 to 50 erbium atoms. The above estimation is unrealistic and can be

the result of using the values of the magnetic moments Mi = gSi instead of the

effective ones, Meff, in estimating the “cluster” composition. The effective moment

of the ErCo2 unit cell containing 16 cobalt and 8 erbium atoms can fit well the

above Meff (cluster) value. Assuming that the effective moment of erbium is given

by its free ion value (9.59 B) and that of cobalt, in this temperature range, of


0.66B, as already mentioned, the corresponding effective moment of the unit cell

is Meff (cell) 27 B, supporting our conclusion. Generally, the effective moments

in characterizing the systems containing atoms with different magnetic

contributions has not a physical significance. The corresponding magnetic

behaviour can be better described by the addition law of the magnetic

susceptibilities, Curie constants, respectively than of the squares of the effective

moments.

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ON THE MAGNETIC BEHAVIOUR OF HEAVY RARE-EARTHS RCo … · energies balance, while the magnetic...

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