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Quadrupolar interactions in rare earth intermetallics
255
Journal of Magnetism and Magnetic Materials 84 (1990) 255-263 North-Holland
QUADRUPOLAR
R. ALBONARD
INTERACTIONS
IN RARE EARTH INTERMETALLICS
and P. MORIN
Laboratoire Louis N6el *, C.N.R.S.,
B.P. 166 X, 38042 Grenoble Cedex, France
A review of quadrupolar interactions in rare earth intermetallic compounds is proposed. Their double nature, magnetoelasticity and pair interactions, is emphasized as well as the role of magnetoelasticity as experimental probe with regard to pair interactions. Their effects on the physical properties of magnetically ordered systems are then discussed in particular in cubic symmetry. Indeed, in rare earth hexaborides, pnictides as well as in CsCl- and AuCus-type compounds, the magnetic properties appear to be deeply determined by quadrupolar couplings.
1. Introduction
In the recent past, multipolar interactions have been extensively analyzed in rare earth intermetallic compounds [l]. These studies concern the complex coexistence of both one-ion (crystalline electric field, CEF) and two-ion interactions. Quadrupolar couplings in particular have been often observed strong enough to compete with the Heisenberg interactions, as it is evidenced by the occurrence of quadrupolar ordering in the paramagnetic range of several compounds. Even when they are dominated by the bilinear interactions, they play a decisive role in the minimization of the free energy of the 4f ions system. Their determination then appears as an inevitable task in numerous rare earth series. For that, the most appropriate experimental probe is the one-ion magnetoelasticity. Our present goal is to discuss some experimental evidences of quadrupolar interactions in rare earth intermetallics and to set a special emphasis on their consequences on the magnetic properties.
Except in the presence of very large bilinear interactions, any classical analysis fails and a quantum treatment is required, which describes the 4f shell properties through spin and multipolar moments within the mean field approximation (MFA) and the Stevens operator formalism [2]. In the following, we will only consider pair interactions between spins and quadrupoles, which have been the most extensively sounded among the high order multipoles. The appropriate Hamiltonian, &‘, which determines the 4f wave functions, includes several terms. The first one is the CEF Hamiltonian X CEF
-
Cq/I~
(1)
P
2. Formalism
where the A,‘s are the CEF parameters and the 4’s the Stevens operators. The numbers and the detailed form of the 4’s depend on the point symmetry of the rare earth site under consideration. The next term, which usually takes place in 2, involves the total angular momentum J through the Zeeman coupling and the isotropic bilinear Heisenberg-type Hamiltonian:
2. I. The Hamiltonian
% = -gJpB(H+
The unquenched orbital moments of the rare earth ions lead to anisotropic magnetic properties.
n = 0*/C is the bilinear exchange coefficient and M = gJpB( J). In this reduced status, this Hamiltonian has failed in the description of the magnetic properties of several series of intermetallics
* Associated with the UniversitC Joseph Fourier, Grenoble 03048853/~/$03.50
0 1990 - Elsevier Science Publishers B.V. (North-Holland)
ng_+aM)
J.
(2)
256
R AIPonard P. Morin / Quadrupolar interactions in rare earth intermetallics
in particular for CsCl- and NaCl-type structure compounds. Indeed the CEF may be deeply modulated by magnetoelastic strains according to the following terms:
variables eB, Q,, as well as H and M: for a given symmetry:
-
xl”( BPcLE, + K’Q,)( H + nM)’
- gy In this expression, the OP2’s represent proper combinations of the second-order Stevens operators, which are coupled with the corresponding symmetrized strain c,, through the magnetoelastic coefficients Bf‘. Anharmonicity as well as two-ion magnetoelasticity are here neglected. This latter hypothesis is usually valid for the majority of the non-S systems. As the Zeeman coupling with respect of the Heisenberg one, the magnetoelastic term is the most appropriate probe for sounding quadrupolar pair interactions:
so= - ~KpQ,Pp, s
with QP = (O,,,). Due to the MFA and equilibrium conditions, two- and one-ion quadrupolar coefficients may lead to a total coefficient, G” = ( BP2/c/j) + KP. co”is the lattice elastic constant in the absence of any 4f shell contribution.
H +
r&q4 + $& + +rN2
+ :K’Q,~.
(5)
The various susceptibilities are only determined by the CEF and, reciprocally, provide us with information about the magnetic and quadrupolar nature of the low-lying levels. x0 is the usual magnetic susceptibility, x,, the strain one; in the presence of a magnetic field, $’ characterizes the quadrupolar response and xy the anisotropic initial curvature of the magnetization curve.
2.3. Experimental techniques With regard to quadrupolar interactions, the most direct experimental probe are the elastic constants through their magnetoelastic softening at low temperatures [3,4]
(6) it is then possible to deduce the absolute value of the magnetoelastic coefficient as well as the
2.2. The free energy For any external stress the free energy may be deduced either by a self-consistent diagonalization of the full Hamiltonian or by applying perturbation methods. The first treatment is relevant for describing magnetic and magnetoelastic properties in the ordered phases, but, due to the great number of parameters simultaneously active, it does not unambiguously demonstrate the relevancy of the basic Hamiltonian. On the contrary, the analytical expansion of the free energy obtained within perturbation methods allows the analysis of several specific experiments. This leads to the determination of the involved parameters, coupling per coupling, symmetry per symmetry. The free energy connects together the associated thermodynamical
14d
CeZn -_--------.-_-_____________
CeMg
-
Fig. 1. Temperature variation of the C., = C,, - C,, mode in CeMg and CeZn. CfI - Cp2 is the lattice behaviour. The full lines are calculated with eq. (6).
R Al.Goonard P. Morin / Quadrupolar interactions in rare earth intermetallics
strength of the pair interactions in the absence of any magnetic term. For instance in the cubic compounds CeMg and CeZn [4], the Cy mode exhibits a pronounced softening down to the first-order antiferromagnetic transition (fig. 1). These large magnetoelastic effects are premonitory evidences of the huge tetragonal strains which develop themselves below TN: c/a = 1 reaches 1.3 and 1.7% in CeMg and CeZn, respectively, at 4.2 K. In the parastriction process [5], the strain is driven by the applied magnetic field renormalized by the bilinear interactions, through the quadrupolar field-susceptibility xr) which is itself renormalized by the quadrupolar pair interactions: 5 =
5
(1
-x$xp) _l:,r*
(7)
(1
This expression may be linearized by plotting as a function of temperature; the slope H/m leads to the magnetoelastic coefficient BP and the shift from (x, (2)) -‘I2 to the total coefficient G,, as it is well demonstrated in CeAg [6] (fig. 2). The last specific experiment is the third order magnetic susceptibility, i.e., the initial curvature of the magnetic response to the applied field [7]. Its CEF contribution, xs), is anisotropic and usually negative. The quadrupolar contribution also de-
CeAg
5-r,“9oYr’
-4
I
257
I
I
20
40 T(K)
Fig. 3. Temperature variation of the third-order magnetic susceptibility in CeMg (trigonal and tetragonal symmetries) and in CeZn (tetragonal symmetry). Note that quadrupolar interactions of trigonal symmetry are not active in the case of a rs ground state.
pends on the field direction. symmetry: x$&, = xh3’ + 2Gfi(~f’)2(1
Thus for a given
- G”&-‘.
According to the values of GP and of the different susceptibilities x$?& may become positive. As shown in fig. 3 for CeZn and CeMg, the initial curvature of the magnetization curve is then described after renormalization by the bilinear coupling:
HXCCNNI [~&)l+1030
TQ
____ :
--: -:
flKI
$(mK)
;
RO
0
Fig. 2. Temperature variation of H/
80 1A,, - XI
(l/’
in CeAg;
h,, - XI = geY is the tetragonal strain induced by the magnetic field H in the paramagnetic phase. Data are described with the parameters indicated for the bilinear and quadrupolar interactions.
x$ is always negative for H parallel to [l 1 11, but is positive for the fourfold symmetry due to the large quadrupolar terms. This set of experimental techniques has led to very coherent determinations for the BP’s and Kp ‘s in the series of TmX (X = Zn, Cd, Ag, Cu, Mg) compounds within the CsCl-type structure and has proved the reliability of the basic Hamiltonian [S]. The coefficients thus achieved have also closely described the occurrence of the ferroquadrupolar orderings as well as the physical properties observed in the ordered phase.
258
R. Alkonard
3. Quadru~lar
P. Morin / Quadrupolar interactions in rare earth intermetallics
40
ordering
The occurrences are concentrated in compounds at each end of the rare earth sequence, where the bilinear interactions are weak according to the De Gennes law and the quadrupolar ones are strong, because those are proportional to the second-order Stevens coefficient. The archetypes of ferroquadrupolar (FQ) systems are TmZn and TmCd, all the properties of which are closely described within the present formalism [9,10]. Quadrupoles order due to pair interactions and the tetragonal strain observed is mainly a consequence of this ordering through the magnetoelastic coupling. Below TQ, the ground state issued from the cubic symmetry E$‘) triplet is nonmagnetic and according to the strength of the bilinear interactions, ferromagnetism can be induced in TmZn, but not in TmCd. CeAg builds a similar case, although a little bit more complex because of the coexistence of FQ interactions with an incipient tendency to lattice instability of electronic origin (6). This latter seems to slightly shift the quadrupolar temperature towards higher values (fig. 4). Antiferroquadrupolar (AFQ) interactions are characterized by negative Kp coefficients. They may drive different types of AFQ structures. First, the lifting of the ground state degeneracy may Table 1 Quadrupolar orderings in rare earth intermetallics. FQ(AFQ) means ferroquadrupolar (antiferroquadrupolar); C, T, t indicate cubic, tetragonal, trigonal symmetries, respectively. In UPd, the transition concerns the U site of nearly cubic symmetry. TQ
6)
Tm+
191
3.12 8.55
CeAg
161
15.85
Ce$
1101
3.3
TmCd [9]
PrPb, [ll] TmGa, [12]
0.35 4.29
PrCu, [13] UW, [14]
1.5 7.5
FQ
AFQ
C-T C-T (T, = 8.12 K) C+T (T, = 5.5 K) C+? (TN = 2.4 K) C+? C+t (TN = 4.26 K) ortho + monoc ?
0’0 4.
‘I
5
10
T(K)
15
20
heat as a function
lead, from one site to the other, to different wave functions, moments and to a ferriquadrupolar structure. Second, changes among the
crystallo-
graphic driven. In both cases, to describe the properties ordered phase one needs a multi-sublattice formalism. From an experimental probes, such as diffraction magnetic field, are required through the existence of different 4f sites. Ce$ [ll], PrPb, [12] and TmGa, [13] are dominated by negative quadrupolar interactions exhibit AFQ orderings.
analysis of the magnetic properties is complicated cerium. PrPb, orders at 0.35 K, a temperature characteristic of very weak AFQ interactions magnetic one has a very narrow existence range. However, magnetic structures multi-q with constant modulus. They indicate
UPd,.
complex are the cases of PrCu, PrCu, is orthorhombic
and
R. AIPonar4 P. Morin / Quadrupolar interactions in rare earth intermetallics
second-order structural transition at 7.5 K [14]. The primary order parameter was found to be the ezx strain. As the CEF is unknown, a quantitative analysis of the observed properties is not possible. UPd, is hexagonal and exhibits properties characteristic of localized magnetism (5f * configuration) [15]. As in Pr metal, there are two U sites, one hexagonal, the other nearly cubic. A well-defined structural transition is observed around 7.5 K. No change of the hexagonal symmetry was observed by X-ray diffraction [16]. However, the increase of the c/a ratio below 7.5 K might be associated with U atoms becoming more prolate at the quasi cubic sites according to an internal ferroquadrupolar process.
4. Effects on the magnetic properties
259
isomorphous CeAg, where bilinear interactions are weak, the ferromagnetic ordering arises only in the FQ phase (fig. 4). The same type of analysis was done in CsCltype TmX: TmCu exhibits a first-order antiferromagnetic transition, whereas TmAg is close to t&criticality, and TmZn and TmCd are quadrupolarly ordered [S]. Among rare earth hexaborides, Pr$ reveals a first order magnetic transition driven by quadrupolar interactions to the same double-q magnetic structure, which also occurs in the AFQ state of Ce$ [18]. In Tb pnictides, the antiferromagnetic ordering is either first-order (TbP, TN = 7.1 K; TbAs, TN = 12.5 K) or second-order (TbSb, TN = 15.5 K; TbBi, TN = 17.5 K). This agrees with positive quadrupolar interactions of trigonal symmetry decreasing from TbP to TbBi [19]; they drive x$, to be positive at TN only in TbP and TbAs as it was observed [20].
4.1. Nature of the magnetic ordering
4.2. Nature of the antiferromagnetic In the majority of compounds the bilinear interactions dominate the quadrupolar ones. The quadrupolar moments, no longer true order parameters, may be expressed as functions of the magnetic ones. The free energy (eq. (5)) transforms into a Landau expansion: F = FCEF+ $I (1 - nx,,)M*
+ ... .
- &r4&$W4 (10)
It allows us to discuss the nature of the magnetic ordering at the disappearance of the M* coefficient [17]. According to the positive (negative) value of x$& (eq. (8)), the transition is first (second) order. This MFA analysis works when magnetization arises either in a ferromagnetic lattice or within each of the antiferromagnetic sublattices. It has successfully explained first-order processes in numerous compounds as soon as the relevant quadrupolar coefficients have been determined in the paramagnetic phase. This is illustrated by CeZn and CeMg; from the values of Gv’s in fig. 3, the FQ orderings would occur at 27 and 12 K, respectively. They are thus hidden by the magnetic ones, but the large quadrupolar terms are responsible of the first-order transitions at TN. Note that in
structure
Below T,, the quadrupolar interactions occur into the determination of the magnetic structure through their contribution to the free energy. In cubic symmetry, the spin structures have the same bilinear energy as soon as they are described by one (collinear arrangements) or more (multiaxial arrangements) propagation vectors of the same star [21]. Thus collinear and multiaxial arrangements are favoured by FQ and AFQ interactions, respectively. In AuCu, and CsCl-type compounds, AF interactions very often coexist with AFQ ones within trigonal symmetry. As soon as the CEF favors (I 1 1) axes, a multiaxial structure is established as in TmGa, [13], DyCu [22], DyAg [23] and NdZn [24]. In the first three compounds, it is triple-q with (f 4 0) propagation vectors. In NdZn, the CEF favors (1 1 1) axes at high temperature, but (1 1 0) ones at low temperature: in both ranges, the structure is multiaxial, triple-q and double-q, respectively, with q belonging to the (i 0 0) star (fig. 5). It seems that, at least in these two large families of compounds, numerous powder neutron diffraction results may be reanalyzed in the spirit. In HOP pnictide, AFQ interactions of tetragonal symmetry coexist with ferromagnetic ones [25].
260
R. AIPonard, P. Morin / Quadrupolar interactions in rare earth intermetallics
Fig. 5. Magnetic structures stabilized by antiferroquadrupolar interactions in antiferromagnetic NdZn and ferromagnetic HOP compounds. The involved quadrupolar interactions have the trigonal and tetragonal symmetry, respectively. In HOP the flopside temperature, TFs, results from the competition of ferromagnetic and antiferroquadrupolar energies. In NdZn the temperature of change of easy direction is determined by both the crystalline electric field and quadrupolar energies.
gether (fig. 6): the lower critical field induces, for the 4 spins which present a component opposite to the applied field, either an inversion along their threefold axis or a jump to another threefold axis. In the first case, the Zeeman energy compensates the bilinear energy variation, AFQ and CEF terms are unmodified. In the second one, the AFQ term also varies. The normalization by the Ntkl temperature values leads to similar critical fields in the three compounds; this indicates the modification to concern mainly the spin energy. The last process towards the forced ferromagnetic phase involves in addition the CEF energy and reveals how energetically unfavoured the fourfold axis is.
HOP orders ferromagnetically at 5.4 K. At 4.8 K, when the AFQ energy is strong enough, a firstorder transition occurs to a flopside state of alternate (1 1 1) ferromagnetic planes. The (0 0 1) axes are always spin axes, but spins change their fourfold direction from a (1 1 1) plane to the next one (fig. 5). The full magnetic phase diagram was analyzed using (positive) bilinear and (negative) quadrupolar coefficients obtained by inelastic neutron scattering in the flopside structure. 4.3. The magnetization structures
processes
in the multiaxial
In the multiaxial structures, the local anisotropy is very large and the magnetization processes are characterized, at least at low temperature, by well-defined magnetization jumps. The three compounds DyAg, DyCu and TmGa, constitute a coherent example; they present the same spontaneous triple-q structure, but very different bilinear interactions (TN = 56.5, 62.8 and 4.21 K, respectively). In spite of this latter difference, their magnetization processes, studied on single crystals by neutron diffraction and magnetic measurements, are very similar and may be analyzed to-
H (kCk) Fig. 6. Field-breaking of the spontaneous multiaxial structure (triple-q), stabilized by antiferroquadrupolar interactions in DyAg, DyCu and TmGas. F defines the forced ferromagnetic phase. The field scale in TmGa, is normalized to the DyAg one according to the ratio of the N&.1 temperatures (Lr, = 56.5 and 4.21 K in DyAg and TmGa,).
261
R. AlPonard P. Morin / Quadrupolar interactions in rare earth intermetaliics 80
According to the sign of the quadrupolar interactions, the first-order transition is shifted towards lower or higher temperature. The modulated structure, which for instance exists at any temperature in HoAg, is confined in DyAg in a small temperature range close to TN, by the trigonal AFQ interactions which quickly stabilize the triple-q structure [23]. This appears as a common feature, observed in other series such as in RI& [18], RCu [27] and AuCu,-type systems [28,29].
3
E
5
-00
u”
60
60
0
I
I
I
100
T(K)
I
200
Fig. 7. Specific heat in NdCd showing the change of easy magnetization direction at 62.5 K. This temperature is in part determined by quadrupolar interactions (inset).
4.4.Magnetic
transitions
in
the ordered phase
Observing changes in the easy magnetization direction is a rather common feature in cubic rare earth intermetallics. Such a first-order transition occurs for instance in Nd, HOC%, Nd, Tb, HoZn, Nd, TbCd and HoAl, ferromagnets and in the NdZn antiferromagnet. It is associated to entropy effects resulting from the anisotropic splitting of the CEF level scheme by the bilinear interactions. However, the quadrupolar terms deeply modify the temperature at which the anisotropy of the free energy changes in sign between axes as in NdCd (fig. 7). A border case of change of easy magnetization axis is set by DyCd and DyZn [22]. As in the isomorphous DyCu and DyAg, the CEF favours (1 1 1) easy axes, but the tetragonal FQ interactions reverse the anisotropy of energy and define the [0 0 l] axis as easy direction in the whole ferromagnetic range. Another example of first-order modification of the magnetic structure is the commensurate-incommensurate transition. Indeed in cubic symmetry, the nature of bilinear interactions often leads to modulated magnetic structures, described by an incommensurate wave vector. In the case of a magnetic ground state, this structure cannot be stable at 0 K due to entropy effects [26] and usually transforms into a commensurate one with constant modulus for spins and thus quadrupoles.
5. Discussion
A coherent determination of the quadrupolar parameters has been achieved in numerous cases, in particular for each of the TmX compounds of CsCl-type [S]. Across a series, the magnetoelastic coefficient keeps the same order of magnitude after the renormalization by the second-order Stevens coefficient (fig. 8), as it was observed for the cubic CEF coefficients for instance in the rare-earth pnictides [30]. As any CEF term, this second-order one, reintroduced in cubic symmetry by the magnetoelasticity, is only partly governed by point charge terms, and also receives an important contribution from the conduction band [311.
CC
Pr
Nd
Tb
Dy
HO
Er
Tm
Fig. 8. Magnetoelastic coefficients for the tetragonal symmetry after normalization to the second order Stevens coefficient in various intermetallic series with the cubic symmetry.
262
R. ALkonard, P. Morin / Quadrupolar interactions in rare earth intemetallics
The pair interaction parameter varies in sign according to the series and to the symmetry: KY and K’ are positive and negative, respectively, in CsCl-type compounds. The opposite situation is observed in NaCl-type ones. In all the compounds quadrupolarly ordered, the pair interactions, positive or a fortiori negative, dominate the (positive) magnetoelastic term and drive the ordering. This is opposite to the case of Jahn-Teller insulators, where magnetoelasticity is the basic mechanism. As the bilinear ones, quadrupolar interactions appear to be mainly mediated by conduction electrons [32]. If the existence of quadrupolar interactions is quite general in rare earth intermetallics, other couplings may also coexist with them. One of them was evoked about CeAg [6], the lattice instability of electronic origin which, however, appears as only minor under normal conditions in CeAg. More general is the magnetoelastic modulation of the bilinear interactions, which occurs through both isotropic and anisotropic modes. Their importance in a given series may be checked in the Gd compound: in GdZn, GdAl, and GdFe, the spontaneous strain is one order of magnitude smaller than in compounds dominated by the CEF [l]. As it is expected to quickly decrease across the series according to the De Gennes law, to neglect it is a good approximation. In pnictides, in particular bismuthides and antimonides, the competition between one- and two-ion strains is more balanced [33,34] and two-ion effects seem to sizeably persist in the series from Gd down to Dy compounds. Save on these special features the two kinds of quadrupolar couplings, magnetoelasticity and pair interactions, are the necessary ingredients for a full understanding of the physical properties observed in rare earth intermetalhcs.
Acknowledgements A large part of our studies on quadrupolar interactions has been achieved in close collaboration with D. Schmitt, E. de Lacheisserie and J. Rouchy. It is a pleasure to thank them for fruitful daily discussions.
References [l] P. Morin and D. Schmitt, in: Handbook on Ferromagnetic Materials, eds. E.P. Wohlfarth and K.H.J. Buschow, vol. 5 (North-Holland, Amsterdam, 1990) p. 30. [2] M.T. Hutchings, Solid State Phys. 16 (1964) 227. [3] J. Kanamori, J. Appl. Phys. 31 (1960) 145. [4] P. Morin, J. Rouchy, T. Nishioka and Y. Miyako, J. Magn. Magn. Mat. 76&77 (1988) 319. [5] P. Morin, D. Schmitt and E. du Tremolet de Lacheisserie, Phys. Rev. B 21 (1980) 1742. [6] P. Morin, J. Magn. Magn. Mat. 71 (1988) 151. [7] P. Morin and D. Schmitt, Phys. Rev. B 23 (1981) 5936. [8] M. Giraud, P. Morin and D. Schmitt, J. Magn. Magn. Mat. 52 (1985) 41. [9] R. AHonard and P. Morin, Phys. Rev. B, 19 (1979) 3868. [lo] D. Givord, P. Morin and D. Schmitt, J. Magn. Magn. Mat. 40 (1983) 121. [ll] J.M. Effantin, J. Rossat-Mignod, P. Burlet, H. Bartholin, S. Kunii and T. Kasuya, J. Magn. Magn. Mat. 47&48 (1985) 145. [12] P. Morin, D. Schmitt and E. du Tremolet de Lacheisserie, J. Magn. Magn. Mat. 30 (1982) 257. [13] P. Morin, M. Giraud, P. Burlet and A. Czopnik, J. Magn. Magn. Mat. 68 (1987) 107. [14] J.K. Kjems, H.R. Ott, S.M. Shapiro and K. Andres, J. de Phys. 39 (1979) C6-1010. [15] K. Andres, D. Davidov, P. Demier, F. Hsu, W.A. Reed and G.J. Nieuwenhuys, Solid State Commun. 28 (1978) 405. [16] P. Zaphnski, D. Meschede, D. Plumacher, W. Schlabitz and H. Schneider, in: Crystalline Electric Field and Structural Effects in f-electron Systems, eds. J.E. Crow, P. Guertin and T.W. Mihahsin (Plenum, New York, 1980) p. 295. 1171 P. Morin and D. Schmitt, Phys. Rev. B 27 (1983) 4412. [18] P. Burlet, J.M. Effantin, J. Rossat-Mignod, S. Kunii and T. Kasuya, J. de Phys. 49 (1988) C8-459. [19] J. Koetzler, Z. Phys. B 55 (1984) 119. [20] G. Raffius and J. Koetzler, Phys. Lett. A 93 (1983) 423. [21] M. Wintemberger and R. Chamard-Bois, Acta Cryst. A28 (1972) 341. [22] R. Al&onard, P. Morin and J. Rouchy, J. Magn. Magn. Mat. 46 (1984) 233. 1231 P. Morin, J. Rouchy, K. Yonenobu, A. Yamagishi and M. Date, J. Magn. Magn. Mat. 81 (1989) 247. [24] P. Morin and A. de Combarieu, Solid State Commun. 17 (1975) 975. [25] P. Fischer, A. Furrer, E. Kaldis, J.K. Kjems and P.M. Levy, Phys. Rev. B 31 (1985) 456. [26] R.J. Elliott, Phys. Rev. 124 (1961) 346. [27] P. Morin and D. Schmitt, J. Magn. Magn. Mat. 21 (1980) 243. (281 P. Morin, M. Giraud, P.L. Regnault, E. Roudaut and A. Czopnik, J. Magn. Magn. Mat. 66 (1987) 345. [29] K. Knorr, A. Loidl, B. St&n and J.L. Buevoz, Z. Phys. B 41 (1981) 315.
R. AIPonard, P. Morin / Quadrupolar interactions in rare earth intermetaliics [30] R.J. Birgeneau, E. Bucher, J.P. Maita, L. Passe11 and K.C. Turberfield, Phys. Rev. B 8 (1973) 5345. [31] P. Morin and D. Schmitt, Phys. Rev. B 23 (1981) 2278. [32] D. Schmitt and P.M. Levy, J. Map Map Mat. 49 (1985) 15.
263
[33] F. HuIIiger and F. Stucki, in: Rare Earth and Actinides, Inst. Phys. Conf. Ser. No. 37 (1978) 92. [34] F. HuIIiger and F. Stucki, Z. Phys. B 31 (1978) 391.
Journal of Magnetism and Magnetic Materials 84 (1990) 255-263 North-Holland
QUADRUPOLAR
R. ALBONARD
INTERACTIONS
IN RARE EARTH INTERMETALLICS
and P. MORIN
Laboratoire Louis N6el *, C.N.R.S.,
B.P. 166 X, 38042 Grenoble Cedex, France
A review of quadrupolar interactions in rare earth intermetallic compounds is proposed. Their double nature, magnetoelasticity and pair interactions, is emphasized as well as the role of magnetoelasticity as experimental probe with regard to pair interactions. Their effects on the physical properties of magnetically ordered systems are then discussed in particular in cubic symmetry. Indeed, in rare earth hexaborides, pnictides as well as in CsCl- and AuCus-type compounds, the magnetic properties appear to be deeply determined by quadrupolar couplings.
1. Introduction
In the recent past, multipolar interactions have been extensively analyzed in rare earth intermetallic compounds [l]. These studies concern the complex coexistence of both one-ion (crystalline electric field, CEF) and two-ion interactions. Quadrupolar couplings in particular have been often observed strong enough to compete with the Heisenberg interactions, as it is evidenced by the occurrence of quadrupolar ordering in the paramagnetic range of several compounds. Even when they are dominated by the bilinear interactions, they play a decisive role in the minimization of the free energy of the 4f ions system. Their determination then appears as an inevitable task in numerous rare earth series. For that, the most appropriate experimental probe is the one-ion magnetoelasticity. Our present goal is to discuss some experimental evidences of quadrupolar interactions in rare earth intermetallics and to set a special emphasis on their consequences on the magnetic properties.
Except in the presence of very large bilinear interactions, any classical analysis fails and a quantum treatment is required, which describes the 4f shell properties through spin and multipolar moments within the mean field approximation (MFA) and the Stevens operator formalism [2]. In the following, we will only consider pair interactions between spins and quadrupoles, which have been the most extensively sounded among the high order multipoles. The appropriate Hamiltonian, &‘, which determines the 4f wave functions, includes several terms. The first one is the CEF Hamiltonian X CEF
-
Cq/I~
(1)
P
2. Formalism
where the A,‘s are the CEF parameters and the 4’s the Stevens operators. The numbers and the detailed form of the 4’s depend on the point symmetry of the rare earth site under consideration. The next term, which usually takes place in 2, involves the total angular momentum J through the Zeeman coupling and the isotropic bilinear Heisenberg-type Hamiltonian:
2. I. The Hamiltonian
% = -gJpB(H+
The unquenched orbital moments of the rare earth ions lead to anisotropic magnetic properties.
n = 0*/C is the bilinear exchange coefficient and M = gJpB( J). In this reduced status, this Hamiltonian has failed in the description of the magnetic properties of several series of intermetallics
* Associated with the UniversitC Joseph Fourier, Grenoble 03048853/~/$03.50
0 1990 - Elsevier Science Publishers B.V. (North-Holland)
ng_+aM)
J.
(2)
256
R AIPonard P. Morin / Quadrupolar interactions in rare earth intermetallics
in particular for CsCl- and NaCl-type structure compounds. Indeed the CEF may be deeply modulated by magnetoelastic strains according to the following terms:
variables eB, Q,, as well as H and M: for a given symmetry:
-
xl”( BPcLE, + K’Q,)( H + nM)’
- gy In this expression, the OP2’s represent proper combinations of the second-order Stevens operators, which are coupled with the corresponding symmetrized strain c,, through the magnetoelastic coefficients Bf‘. Anharmonicity as well as two-ion magnetoelasticity are here neglected. This latter hypothesis is usually valid for the majority of the non-S systems. As the Zeeman coupling with respect of the Heisenberg one, the magnetoelastic term is the most appropriate probe for sounding quadrupolar pair interactions:
so= - ~KpQ,Pp, s
with QP = (O,,,). Due to the MFA and equilibrium conditions, two- and one-ion quadrupolar coefficients may lead to a total coefficient, G” = ( BP2/c/j) + KP. co”is the lattice elastic constant in the absence of any 4f shell contribution.
H +
r&q4 + $& + +rN2
+ :K’Q,~.
(5)
The various susceptibilities are only determined by the CEF and, reciprocally, provide us with information about the magnetic and quadrupolar nature of the low-lying levels. x0 is the usual magnetic susceptibility, x,, the strain one; in the presence of a magnetic field, $’ characterizes the quadrupolar response and xy the anisotropic initial curvature of the magnetization curve.
2.3. Experimental techniques With regard to quadrupolar interactions, the most direct experimental probe are the elastic constants through their magnetoelastic softening at low temperatures [3,4]
(6) it is then possible to deduce the absolute value of the magnetoelastic coefficient as well as the
2.2. The free energy For any external stress the free energy may be deduced either by a self-consistent diagonalization of the full Hamiltonian or by applying perturbation methods. The first treatment is relevant for describing magnetic and magnetoelastic properties in the ordered phases, but, due to the great number of parameters simultaneously active, it does not unambiguously demonstrate the relevancy of the basic Hamiltonian. On the contrary, the analytical expansion of the free energy obtained within perturbation methods allows the analysis of several specific experiments. This leads to the determination of the involved parameters, coupling per coupling, symmetry per symmetry. The free energy connects together the associated thermodynamical
14d
CeZn -_--------.-_-_____________
CeMg
-
Fig. 1. Temperature variation of the C., = C,, - C,, mode in CeMg and CeZn. CfI - Cp2 is the lattice behaviour. The full lines are calculated with eq. (6).
R Al.Goonard P. Morin / Quadrupolar interactions in rare earth intermetallics
strength of the pair interactions in the absence of any magnetic term. For instance in the cubic compounds CeMg and CeZn [4], the Cy mode exhibits a pronounced softening down to the first-order antiferromagnetic transition (fig. 1). These large magnetoelastic effects are premonitory evidences of the huge tetragonal strains which develop themselves below TN: c/a = 1 reaches 1.3 and 1.7% in CeMg and CeZn, respectively, at 4.2 K. In the parastriction process [5], the strain is driven by the applied magnetic field renormalized by the bilinear interactions, through the quadrupolar field-susceptibility xr) which is itself renormalized by the quadrupolar pair interactions: 5 =
5
(1
-x$xp) _l:,r*
(7)
(1
This expression may be linearized by plotting as a function of temperature; the slope H/m leads to the magnetoelastic coefficient BP and the shift from (x, (2)) -‘I2 to the total coefficient G,, as it is well demonstrated in CeAg [6] (fig. 2). The last specific experiment is the third order magnetic susceptibility, i.e., the initial curvature of the magnetic response to the applied field [7]. Its CEF contribution, xs), is anisotropic and usually negative. The quadrupolar contribution also de-
CeAg
5-r,“9oYr’
-4
I
257
I
I
20
40 T(K)
Fig. 3. Temperature variation of the third-order magnetic susceptibility in CeMg (trigonal and tetragonal symmetries) and in CeZn (tetragonal symmetry). Note that quadrupolar interactions of trigonal symmetry are not active in the case of a rs ground state.
pends on the field direction. symmetry: x$&, = xh3’ + 2Gfi(~f’)2(1
Thus for a given
- G”&-‘.
According to the values of GP and of the different susceptibilities x$?& may become positive. As shown in fig. 3 for CeZn and CeMg, the initial curvature of the magnetization curve is then described after renormalization by the bilinear coupling:
HXCCNNI [~&)l+1030
TQ
____ :
--: -:
flKI
$(mK)
;
RO
0
Fig. 2. Temperature variation of H/
80 1A,, - XI
(l/’
in CeAg;
h,, - XI = geY is the tetragonal strain induced by the magnetic field H in the paramagnetic phase. Data are described with the parameters indicated for the bilinear and quadrupolar interactions.
x$ is always negative for H parallel to [l 1 11, but is positive for the fourfold symmetry due to the large quadrupolar terms. This set of experimental techniques has led to very coherent determinations for the BP’s and Kp ‘s in the series of TmX (X = Zn, Cd, Ag, Cu, Mg) compounds within the CsCl-type structure and has proved the reliability of the basic Hamiltonian [S]. The coefficients thus achieved have also closely described the occurrence of the ferroquadrupolar orderings as well as the physical properties observed in the ordered phase.
258
R. Alkonard
3. Quadru~lar
P. Morin / Quadrupolar interactions in rare earth intermetallics
40
ordering
The occurrences are concentrated in compounds at each end of the rare earth sequence, where the bilinear interactions are weak according to the De Gennes law and the quadrupolar ones are strong, because those are proportional to the second-order Stevens coefficient. The archetypes of ferroquadrupolar (FQ) systems are TmZn and TmCd, all the properties of which are closely described within the present formalism [9,10]. Quadrupoles order due to pair interactions and the tetragonal strain observed is mainly a consequence of this ordering through the magnetoelastic coupling. Below TQ, the ground state issued from the cubic symmetry E$‘) triplet is nonmagnetic and according to the strength of the bilinear interactions, ferromagnetism can be induced in TmZn, but not in TmCd. CeAg builds a similar case, although a little bit more complex because of the coexistence of FQ interactions with an incipient tendency to lattice instability of electronic origin (6). This latter seems to slightly shift the quadrupolar temperature towards higher values (fig. 4). Antiferroquadrupolar (AFQ) interactions are characterized by negative Kp coefficients. They may drive different types of AFQ structures. First, the lifting of the ground state degeneracy may Table 1 Quadrupolar orderings in rare earth intermetallics. FQ(AFQ) means ferroquadrupolar (antiferroquadrupolar); C, T, t indicate cubic, tetragonal, trigonal symmetries, respectively. In UPd, the transition concerns the U site of nearly cubic symmetry. TQ
6)
Tm+
191
3.12 8.55
CeAg
161
15.85
Ce$
1101
3.3
TmCd [9]
PrPb, [ll] TmGa, [12]
0.35 4.29
PrCu, [13] UW, [14]
1.5 7.5
FQ
AFQ
C-T C-T (T, = 8.12 K) C+T (T, = 5.5 K) C+? (TN = 2.4 K) C+? C+t (TN = 4.26 K) ortho + monoc ?
0’0 4.
‘I
5
10
T(K)
15
20
heat as a function
lead, from one site to the other, to different wave functions, moments and to a ferriquadrupolar structure. Second, changes among the
crystallo-
graphic driven. In both cases, to describe the properties ordered phase one needs a multi-sublattice formalism. From an experimental probes, such as diffraction magnetic field, are required through the existence of different 4f sites. Ce$ [ll], PrPb, [12] and TmGa, [13] are dominated by negative quadrupolar interactions exhibit AFQ orderings.
analysis of the magnetic properties is complicated cerium. PrPb, orders at 0.35 K, a temperature characteristic of very weak AFQ interactions magnetic one has a very narrow existence range. However, magnetic structures multi-q with constant modulus. They indicate
UPd,.
complex are the cases of PrCu, PrCu, is orthorhombic
and
R. AIPonar4 P. Morin / Quadrupolar interactions in rare earth intermetallics
second-order structural transition at 7.5 K [14]. The primary order parameter was found to be the ezx strain. As the CEF is unknown, a quantitative analysis of the observed properties is not possible. UPd, is hexagonal and exhibits properties characteristic of localized magnetism (5f * configuration) [15]. As in Pr metal, there are two U sites, one hexagonal, the other nearly cubic. A well-defined structural transition is observed around 7.5 K. No change of the hexagonal symmetry was observed by X-ray diffraction [16]. However, the increase of the c/a ratio below 7.5 K might be associated with U atoms becoming more prolate at the quasi cubic sites according to an internal ferroquadrupolar process.
4. Effects on the magnetic properties
259
isomorphous CeAg, where bilinear interactions are weak, the ferromagnetic ordering arises only in the FQ phase (fig. 4). The same type of analysis was done in CsCltype TmX: TmCu exhibits a first-order antiferromagnetic transition, whereas TmAg is close to t&criticality, and TmZn and TmCd are quadrupolarly ordered [S]. Among rare earth hexaborides, Pr$ reveals a first order magnetic transition driven by quadrupolar interactions to the same double-q magnetic structure, which also occurs in the AFQ state of Ce$ [18]. In Tb pnictides, the antiferromagnetic ordering is either first-order (TbP, TN = 7.1 K; TbAs, TN = 12.5 K) or second-order (TbSb, TN = 15.5 K; TbBi, TN = 17.5 K). This agrees with positive quadrupolar interactions of trigonal symmetry decreasing from TbP to TbBi [19]; they drive x$, to be positive at TN only in TbP and TbAs as it was observed [20].
4.1. Nature of the magnetic ordering
4.2. Nature of the antiferromagnetic In the majority of compounds the bilinear interactions dominate the quadrupolar ones. The quadrupolar moments, no longer true order parameters, may be expressed as functions of the magnetic ones. The free energy (eq. (5)) transforms into a Landau expansion: F = FCEF+ $I (1 - nx,,)M*
+ ... .
- &r4&$W4 (10)
It allows us to discuss the nature of the magnetic ordering at the disappearance of the M* coefficient [17]. According to the positive (negative) value of x$& (eq. (8)), the transition is first (second) order. This MFA analysis works when magnetization arises either in a ferromagnetic lattice or within each of the antiferromagnetic sublattices. It has successfully explained first-order processes in numerous compounds as soon as the relevant quadrupolar coefficients have been determined in the paramagnetic phase. This is illustrated by CeZn and CeMg; from the values of Gv’s in fig. 3, the FQ orderings would occur at 27 and 12 K, respectively. They are thus hidden by the magnetic ones, but the large quadrupolar terms are responsible of the first-order transitions at TN. Note that in
structure
Below T,, the quadrupolar interactions occur into the determination of the magnetic structure through their contribution to the free energy. In cubic symmetry, the spin structures have the same bilinear energy as soon as they are described by one (collinear arrangements) or more (multiaxial arrangements) propagation vectors of the same star [21]. Thus collinear and multiaxial arrangements are favoured by FQ and AFQ interactions, respectively. In AuCu, and CsCl-type compounds, AF interactions very often coexist with AFQ ones within trigonal symmetry. As soon as the CEF favors (I 1 1) axes, a multiaxial structure is established as in TmGa, [13], DyCu [22], DyAg [23] and NdZn [24]. In the first three compounds, it is triple-q with (f 4 0) propagation vectors. In NdZn, the CEF favors (1 1 1) axes at high temperature, but (1 1 0) ones at low temperature: in both ranges, the structure is multiaxial, triple-q and double-q, respectively, with q belonging to the (i 0 0) star (fig. 5). It seems that, at least in these two large families of compounds, numerous powder neutron diffraction results may be reanalyzed in the spirit. In HOP pnictide, AFQ interactions of tetragonal symmetry coexist with ferromagnetic ones [25].
260
R. AIPonard, P. Morin / Quadrupolar interactions in rare earth intermetallics
Fig. 5. Magnetic structures stabilized by antiferroquadrupolar interactions in antiferromagnetic NdZn and ferromagnetic HOP compounds. The involved quadrupolar interactions have the trigonal and tetragonal symmetry, respectively. In HOP the flopside temperature, TFs, results from the competition of ferromagnetic and antiferroquadrupolar energies. In NdZn the temperature of change of easy direction is determined by both the crystalline electric field and quadrupolar energies.
gether (fig. 6): the lower critical field induces, for the 4 spins which present a component opposite to the applied field, either an inversion along their threefold axis or a jump to another threefold axis. In the first case, the Zeeman energy compensates the bilinear energy variation, AFQ and CEF terms are unmodified. In the second one, the AFQ term also varies. The normalization by the Ntkl temperature values leads to similar critical fields in the three compounds; this indicates the modification to concern mainly the spin energy. The last process towards the forced ferromagnetic phase involves in addition the CEF energy and reveals how energetically unfavoured the fourfold axis is.
HOP orders ferromagnetically at 5.4 K. At 4.8 K, when the AFQ energy is strong enough, a firstorder transition occurs to a flopside state of alternate (1 1 1) ferromagnetic planes. The (0 0 1) axes are always spin axes, but spins change their fourfold direction from a (1 1 1) plane to the next one (fig. 5). The full magnetic phase diagram was analyzed using (positive) bilinear and (negative) quadrupolar coefficients obtained by inelastic neutron scattering in the flopside structure. 4.3. The magnetization structures
processes
in the multiaxial
In the multiaxial structures, the local anisotropy is very large and the magnetization processes are characterized, at least at low temperature, by well-defined magnetization jumps. The three compounds DyAg, DyCu and TmGa, constitute a coherent example; they present the same spontaneous triple-q structure, but very different bilinear interactions (TN = 56.5, 62.8 and 4.21 K, respectively). In spite of this latter difference, their magnetization processes, studied on single crystals by neutron diffraction and magnetic measurements, are very similar and may be analyzed to-
H (kCk) Fig. 6. Field-breaking of the spontaneous multiaxial structure (triple-q), stabilized by antiferroquadrupolar interactions in DyAg, DyCu and TmGas. F defines the forced ferromagnetic phase. The field scale in TmGa, is normalized to the DyAg one according to the ratio of the N&.1 temperatures (Lr, = 56.5 and 4.21 K in DyAg and TmGa,).
261
R. AlPonard P. Morin / Quadrupolar interactions in rare earth intermetaliics 80
According to the sign of the quadrupolar interactions, the first-order transition is shifted towards lower or higher temperature. The modulated structure, which for instance exists at any temperature in HoAg, is confined in DyAg in a small temperature range close to TN, by the trigonal AFQ interactions which quickly stabilize the triple-q structure [23]. This appears as a common feature, observed in other series such as in RI& [18], RCu [27] and AuCu,-type systems [28,29].
3
E
5
-00
u”
60
60
0
I
I
I
100
T(K)
I
200
Fig. 7. Specific heat in NdCd showing the change of easy magnetization direction at 62.5 K. This temperature is in part determined by quadrupolar interactions (inset).
4.4.Magnetic
transitions
in
the ordered phase
Observing changes in the easy magnetization direction is a rather common feature in cubic rare earth intermetallics. Such a first-order transition occurs for instance in Nd, HOC%, Nd, Tb, HoZn, Nd, TbCd and HoAl, ferromagnets and in the NdZn antiferromagnet. It is associated to entropy effects resulting from the anisotropic splitting of the CEF level scheme by the bilinear interactions. However, the quadrupolar terms deeply modify the temperature at which the anisotropy of the free energy changes in sign between axes as in NdCd (fig. 7). A border case of change of easy magnetization axis is set by DyCd and DyZn [22]. As in the isomorphous DyCu and DyAg, the CEF favours (1 1 1) easy axes, but the tetragonal FQ interactions reverse the anisotropy of energy and define the [0 0 l] axis as easy direction in the whole ferromagnetic range. Another example of first-order modification of the magnetic structure is the commensurate-incommensurate transition. Indeed in cubic symmetry, the nature of bilinear interactions often leads to modulated magnetic structures, described by an incommensurate wave vector. In the case of a magnetic ground state, this structure cannot be stable at 0 K due to entropy effects [26] and usually transforms into a commensurate one with constant modulus for spins and thus quadrupoles.
5. Discussion
A coherent determination of the quadrupolar parameters has been achieved in numerous cases, in particular for each of the TmX compounds of CsCl-type [S]. Across a series, the magnetoelastic coefficient keeps the same order of magnitude after the renormalization by the second-order Stevens coefficient (fig. 8), as it was observed for the cubic CEF coefficients for instance in the rare-earth pnictides [30]. As any CEF term, this second-order one, reintroduced in cubic symmetry by the magnetoelasticity, is only partly governed by point charge terms, and also receives an important contribution from the conduction band [311.
CC
Pr
Nd
Tb
Dy
HO
Er
Tm
Fig. 8. Magnetoelastic coefficients for the tetragonal symmetry after normalization to the second order Stevens coefficient in various intermetallic series with the cubic symmetry.
262
R. ALkonard, P. Morin / Quadrupolar interactions in rare earth intemetallics
The pair interaction parameter varies in sign according to the series and to the symmetry: KY and K’ are positive and negative, respectively, in CsCl-type compounds. The opposite situation is observed in NaCl-type ones. In all the compounds quadrupolarly ordered, the pair interactions, positive or a fortiori negative, dominate the (positive) magnetoelastic term and drive the ordering. This is opposite to the case of Jahn-Teller insulators, where magnetoelasticity is the basic mechanism. As the bilinear ones, quadrupolar interactions appear to be mainly mediated by conduction electrons [32]. If the existence of quadrupolar interactions is quite general in rare earth intermetallics, other couplings may also coexist with them. One of them was evoked about CeAg [6], the lattice instability of electronic origin which, however, appears as only minor under normal conditions in CeAg. More general is the magnetoelastic modulation of the bilinear interactions, which occurs through both isotropic and anisotropic modes. Their importance in a given series may be checked in the Gd compound: in GdZn, GdAl, and GdFe, the spontaneous strain is one order of magnitude smaller than in compounds dominated by the CEF [l]. As it is expected to quickly decrease across the series according to the De Gennes law, to neglect it is a good approximation. In pnictides, in particular bismuthides and antimonides, the competition between one- and two-ion strains is more balanced [33,34] and two-ion effects seem to sizeably persist in the series from Gd down to Dy compounds. Save on these special features the two kinds of quadrupolar couplings, magnetoelasticity and pair interactions, are the necessary ingredients for a full understanding of the physical properties observed in rare earth intermetalhcs.
Acknowledgements A large part of our studies on quadrupolar interactions has been achieved in close collaboration with D. Schmitt, E. de Lacheisserie and J. Rouchy. It is a pleasure to thank them for fruitful daily discussions.
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