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Interplay of antiferromagnetic and antiferroquadrupolar interactions in DyAg and other rare earth intermetallic compounds
Journal of Magnetism and Magnetic North-Holland, Amsterdam
Materials
247
81 (1989) 247-258
INTERPLAY OF ANTIFERROMAGNETIC AND ANTIFERROQUADRUPOLAR INTERACTIONS IN DyAg AND OTHER RARE EARTH INTERMETALLIC COMPOUNDS P. MORIN,
J. ROUCHY,
K. YONENOBU
a, A. YAMAGISHI
a and M. DATE
a
Laboratoire Louis Nt!el ‘, CNRS, 166 X, 38042 Grenoble Cedex, France ’ Research Center for Extreme Materials, Osaka University, Toyonaka, Osaka 560, Japan
Received
25 May 1989; in revised form 11 July 1989
The study of the magnetoelasticity of the cubic (CsCl-type) rare earth intermetallic DyAg allows us to determine the strength of both the magnetoelastic coupling and the quadrupolar pair interactions. These latter ones are observed to be negative (antiferroquadrupolar type) for the tetragonal symmetry as well as for the trigonal one. They drive the magnetic structure to be triple-q at low temperature: the cubic magnetic cell consists of four pairs of ferromagnetic moments pointing along each of the treefold axes. At high temperature, it is replaced by a double-q structure, then, immediately below TN, by a modulated arrangement. The magnetization processes have been thoroughly studied along the three main axes in fields up to 40 T and compared with previous results in isomorphous DyCu and in the AuCu,-type compound TmGa,. The sequences under field of the different magnetic structures are identical and mainly determined by the crytalline electric field and the antiferroquadrupolar interactions. These 3 compounds do not set a peculiar case, but seem to belong to a larger family of cubic compounds with multiaxial structures governed by antiferroquadrupolar terms.
1. Introduction The intermetallic equiatomic rare earth-silver compounds crystallize within the C&l-type cubic structure. This simple arrangement and the nonmagnetic alloyed metal provide us with favourable conditions to thoroughly study the physical properties associated with the 4f shell. The RAg series is characterized by negative bilinear interactions. At least close to the NCel temperature, the actual magnetic structure is defined by a propagation vector which is not commensurate with the lattice cell. Indeed, modulated magnetic structures have been observed in TmAg [l], ErAg [2], HoAg [1,3]. The propagation vector is (i - 7, f, 0) with r around 0.07 and exhibits a weak temperature dependence [l]. In ErAg, a first-order transition at 9.5 K separates this modulated structure from the commensurate ($ f 0) one, stable at low temperature. Similar situations have been observed in isomorphous RCu compounds [4]. ’ Associated
with the Universite
Joseph
Fourier,
Grenoble.
0304-8853/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
In DyAg, two different ordered phases have been recently observed by both magnetic susceptibility and powder neutron diffraction measurements [5]. At low temperature, the (1 i 0) propagation vector previously determined by Arnold [6] is confirmed; between 46.5 K and TN = 56.6 K, a modulated structure is established in a way reminiscent of the ErAg situation. The presence of only two equivalent satellites for the (: : 0) position on the powder neutron diffraction pattern at 50 K (fig. 2c in ref. [5]) indicates that the propagation vector is (f - 7, i, 0) as in other RAg compounds. At low temperatures, the neutron diffraction pattern does not simply indicate that the magnetic structure is collinear. Indeed multiaxis (non-collinear) arrangements are possible, which are described by the presence of several propagation vectors from the (i t 0) star in each magnetic domain [7]. In the case of DyAg as for DyCu [4,8], the collinear (single-q) structure with magnetic moments along one fourfold axis is in competition with one double-q structure and one triple-q B.V.
248
P. Mm-in et al. / Interplay of interactions
structure (magnetic moments along two twofold axes and four threefold axes, respectively, as drawn in fig. 1 of ref. [4]). The lattice symmetry driven by the magnetic system through the magnetoelastic coupling is tetragonal in the case of the collinear and biaxial structures, but remains cubic in the case of the triple-q one. Experimentally, no lattice distortion has been detected in DyAg and DyCu by either neutron diffraction [5,6] or X-ray diffraction [9]. Our present aim is to show by means of parastriction and sound velocity measurements that the absence of any cell distortion is incompatible with the strength of the magnetoelastic coupling and only results from the cubic symmetry of the triple-q magnetic structure. We also want to establish that in spite of the intrinsic nature of the bilinear interactions which favours incommensurability, this triple-q structure is stabilized at low temperature by quadrupolar interactions, negative within the trigonal symmetry. Last, the study of the magnetic phase diagrams under a magnetic field will be presented and discussed in comparison with analogous diagrams obtained in other compounds such as DyCu [lo] and TmGa s [ll] which exhibit the same non-collinear spontaneous magnetic structure.
in DyAg
about 14 K. The other levels are at about 155 K. This level scheme favors a threefold axis as easy magnetization direction. The term: xo
= -&P&g_&J)
+ H)J
(3)
includes the Zeeman coupling in the external field H and the Heisenberg-type exchange written in the mean field approximation. From the exchange coefficient n = 0*/C determined by the temperature dependence of the magnetic susceptibility, a O* value of -27 K is deduced (section 3.3.1). For both tetragonal and trigonal symmetries, the quadrupolar terms have contributions from the magnetoelastic coupling with the lattice and from the pair interactions between rare earth sites: XQ = -((By)2/coy
+ IV)(@)@
-((Bf)2/c~+K~)((P,y)pIv+
+ 3(@)@) ...)?
(4)
BY, B’ and KY, K’ are the magnetoelastic and pair interaction coefficients, respectively. The magnetic properties in the paramagnetic state can be described using single ion susceptibilities, the expression of which are established from the cubic CEF level scheme using a perturbation theory. They allow us to determine the various quadrupolar coefficients by means of independent experimental ways [14,15].
2. Formalism The basic Hamiltonian describing the magnetic properties includes the following single-ion terms [12]:
3. Experiments
SP=.%&r
A magnetic field applied along a fourfold (or a trigonal) axis induces a change of length, which may be analyzed using the strain susceptibility x,(x,), the quadrupolar field-susceptibility xl” (xi”) and the first-order magnetic susceptibility, x0. For instance, for the tetragonal symmetry, the tetragonal strain ey may be analyzed with the following relation, which allows a linearized temperature variation in vanishing CEF effects [15].
+.%?o +xo.
(1)
The form .%cnr = wxo,
+ w( 1 - 1x I) 0,
(2)
describes the Crystalline Electric Field (CEF), using the parametrization of Lea Leask and Wolf [13]. As discussed in ref. [lo], the values W = - 0.48 K and x = -0.55 are coherent with the parameters determined in Ho-, Er- and TmAg isomorphous compounds and are also very close to the DyCu ones. These values will be kept constant in the following. They define a level scheme with the l?d3’ ground state and the I’$‘) first excited level at
3.1. Parastriction
H/p
= /m(l/@) x (1 - GYx$‘*(l
- nx,,),
(5)
249
P. Morin et al. / Interplay of interactions in DyAg
1 0
60
180 120 T(K)
240
I 300 *.
Fig. 1. Temperature dependence of H/@ for the tetragonal symmetry in DyAg. ry is the tetragonal strain. The experimental variation is described with the parameters indicated and using eq. (5) in the text.
3.6
I 0
...
I
100
200
-...
se
300
T(K)
Fig. 2. Temperature variation of the C’ = C, and C” = (C,, + 2C,s)/3 normal elastic modes in DyAg.
C,Y = Cfl - Cfz is the non-magnetic lattice elastic constant, GY = P/CJ + KY the total quadrupolar coefficient. Changes of length induced by magnetic fields up to 10 kOe have been measured along the [OOl] axis of a monocrystalline sphere of 4.7 mm in diameter in a capacitance dilatometer [16]. The data of fig. 1 are described using O* = - 27 K (see section 3.3.1) and /m = 115. With C,Y = 1.615 X lo5 K/at (see section 3.2) the magnetoelastic coefficient follows as BY = + 12 K/at. The fit is slightly improved by introducing weakly negative quadrupolar interactions, characterized by GY around - 10 mK. According to the experimental uncertainties and the interplay between the parameters, BY may range from 12 to 9 K whereas GY varies from - 10 to 0 mK. The total quadrupolar interactions of tetragonal symmetry are small, even slightly negative. As the CEF terms themselves, they do not favor a fourfold axis as the direction of easy magnetization and then forbid the collinear antiferromagnetic structure to occur. The BY value is reminiscent of the parastriction analysis in DyCu [lo] (B, = BY = 15 K) and in DyZn [15] (By = 10 K). The immediate consequence is that in the case of the simple-q magnetic structure or of the double-q one, the tetragonal spontaneous strain is calculated to reach 8 X lop3 (as in DyZn) or - 4 X 10e3, in full contradiction with the zero experimental value. Thus among the three magnetic structures compatible with the low temperatures
neutron diffraction spectrum, the triple-q structure is the actual one. Applying the magnetic field along [ill] leads to changes of length too small to be significantly analyzed. In DyAg as in DyCu and DyZn, the B’ magnetoelastic coupling is very weak. This also agrees with other determinations in isomorphous series [12]. 3.2. Elastic constants Samples with faces (111) planes have been ties corresponding to tions C,, - C12, C,, + and C,, + 2C,, + 4C,
I
parallel to (lOO), (110) and used. The ultrasonic velociC,,, C, and the combinaC12 + 2C44, C,, - C12 + C44 have been measured be-
I
1
u
1
J
t
1.6-
0
100
T (K)
200
300
Fig. 3. Temperature variation of the Cy = (C,, - Cl,)/2 elastic mode in DyAg. Data in the paramagnetic range are described using eq. (6) with the parameters indicated.
250
P. Morin et al. / Interplay of interactions in DyAg
tween 2 and 300 K. These led to the temperature variation of the three cubic normal modes Cy = C,, - C,,, C’ = C, and C” = )(C,, + 2C,,) (figs. 2 and 3). As previously discussed, the absolute values of the three cubic modes at room temperature are characteristics of the stability of the CsCl-type structure in RAg [1’7]. Indeed, it is well known that this arrangement becomes less stable when proceeding up the series and the instability is particularly obvious for La and CeAg. Considering the large values of their elastic constants, the lattices of DyAg and YAg - the present non-magnetic reference - can be estimated as quite stable. For DyAg as for DyCu, all the modes remain observable at low temperatures below TN. This indicates the absence of any sizeable magnetoelastic interaction with magnetic domains. This is in contrast to the case of magnetic compounds tetragonally strained such as TmCu [18], TmAg [19], where the ultrasonic waves, Cy in particular, lose all their energy in vibrating the domain walls. Thus, this ultrasonic technique constitutes quite a relevant probe to determine the existence of a
magnetic triple-q structure which maintains the cubic symmetry. Other fruitful information, which is provided in the ordered range, is the existence of two clear anomalies at T,*= 47.4 K and T,*= 49.0K below the N6el temperature in fig. 4. The lower temperature corresponds to the transition previously observed at 46.5 K [5]. The transition at T2* has not been detected up to now although it is clearly evidenced here. We have observed this pair of transitions in all the different single crystals and annealed polycrystals we have prepared; the analysis of magnetization measurements will show that both of them are intrinsic properties of the compound. The actual value of the Neel temperature, higher than 55 K, is difficult to accurately determine from the temperature variations in fig. 4. This is mainly due to the fact that the second order transition between incommensurate and paramagnetic phases is very smooth; this is associated with the presence of weak magnetic moments on given 4f sites. In the paramagnetic phase, the magnetoelastic contribution to the measured elastic constants is rather weak; the temperature dependence of the bulk modulus is quite normal. This is indicative of the weakness of the cubic magnetoelastic coupling and has been observed in all CsCl-type rare earth compounds [12]. The C’ mode exhibits a linear temperature variation down to 100 K. The small downwards curvature observed above TN cannot be described within the susceptibility formalism and seems to be due to precursory effects associated with magnetic ordering. The temperature dependence of Cy is not linear in contrast with the behaviour observed in YAg. The small softening induced by the magnetoelastic coupling may be described by the usual expression: cy = c,y - PXJ(l
Fig. 4. Temperature variation of the 3 normal elastic modes in the range of the magnetic transitions. The behaviours are very different for the 3 modes, in particular around T,* and 7”*, the temperatures demarcating the commensurate phases.
-KY&).
(6)
The calculated strain susceptibility xv exhibits a Van Vleck behaviour at low temperature, which prevents the magnetoelastic contribution from drastically varying as a function of temperature. Considering the anharmonicity of the lattice through a C,Y background with the same slope as
P. Morin et al. / Interplay of interactions in DyAg
251
in YAg leads to a magnetoelastic coefficient, BY = 5 K (fig. 3). Changing the background slope by a ratio of 2 gives a similar agreement with BY = 11 K. Note that negative KY values could slightly increase BY for a constant C,Y.This weakness of the softening forbids accurate determinations of the quadrupolar parameters although their values are in agreement with their determination by parastriction. 3.3. Magnetization Magnetization processes have been extensively studied for several ranges of the magnetic field and temperatures. First, numerous data have been collected in the Louis NCel Laboratory in fields up to 76 kOe applied along the three main crystallographic directions of a monocrystalline sphere. Through Arrott’s plots as well as field or temperature derivatives, they allow us to determine the strength of the bilinear interactions, the quadrupolar contribution to the third-order magnetic susceptibility and to obtain precise information on the magnetic phase diagrams, in particular around the N6el temperature. 3.3.1. Paramagnetic
phase
Isothermal magnetization curves have been collected between 60 and 100 K. From the extrapolated null magnetization value in Arrott’s plots, the reciprocal magnetic susceptibility was found perfectly isotropic and its temperature variation is drawn in fig. 5. The difference between this exI
60
I
4 Ag
60
70 1 (K)
80
90
Fig. 6. Temperature variation of the third-order magnetic susceptibilities for the trigonal and tetragonal symmetries in DyAg. The different lines are calculated with the parameters indicated according to eq. (7). In both cases, the quadrupolar contributions are negative.
perimental behaviour and the calculated one (l/x = l/x,, - 0*/C) leads to a value 8 = - 27 K, which characterizes the bilinear interactions. The analysis of the slope of Arrott’s plots as well as of M/H (Hz) curves leads to the temperature variation of the initial curvature of the magnetization curve, i.e., of the third-order magnetic susceptibility [14,20]. This latter one was shown to be sensitive to quadrupolar contributions, which modify its usual behaviour described in the CEF model by xl” and x,(3) for the tetragonal and trigonal symmetries, respectively:
xfj = [l - (o*/c)xo] -4 x
[xl”+24 ~3~/(1-G’XI)]
XQ = [l - (o*/c)x,] 301 50
70
90
I
T (K) Fig. 5. Temperature variation of the reciprocal first-order magnetic susceptibility in DyAg. The shift between data and the single-ion behaviour (hatched line) defines the bilinear interactions coefficient, 8 * = - 27 K.
7
/,\
-4
x [ xi3) + 6G’( ~l”)~/(l-
3~7x4.
The very small values, which are observed for both symmetries close to the limit of sensitivity, are due to the large negative bilinear interactions (@* = -27 K) (fig. 6). For the tetragonal symmetry, the
252
P. Morin et al. / Interplay of interactions
in DyAg
data indicate that the quadrupolar interactions are slightly negative. Below 70 K, a small shift between the data and calculated values is observed as previously seen for the temperature variations of the reciprocal first-order susceptibility. It exists also for trigonal symmetry and may be the signature of magnetic fluctuations. For [ill], the quadrupolar contribution is obviously negative, as systematically observed in CsCl-type rare earth intermetallics [12].
... :. : . : : : .. . .
.*...‘..
45.2OK
3.3.2. Magnetic phase diagrams in low fields The determination of the NCel temperature was attempted by means of Arrott’s plots of isothermal magnetization curves in fields applied along [OOl], [Oil] and [ill] (fig. 7). At high temperatures, Arrott plots are linear with a positive slope (negative curvature for M(H)). At low temperature, the slope is negative in the range where the magnetization is small (positive curvature for M(H)). The N6el temperature is defined by the infinite slope of the corresponding Arrott plot. Thus TN lies in the range (56.3, 56.5 K). This result was confirmed by analyzing also the field derivatives (fig. 8). At TN, dM/dH is con-
4
.... . . .. . . *
2.)
. .. ..
$
.
...’ ?.
.... 27*............i.........
‘..‘....‘...,.
.‘..
2.“
.,,...............*
. .. . .._
5?.BOK :_. /..:.* ......:. ....:. :’
t .* ..‘... i
t
25l 0
20
H( kOe)
I
60
40
Fig. 8. Field derivative of the [OOl] isothermal magnetization the temperature range of the NCel transition of DyAg.
in
stant in a large range. Below TN, a smooth maximum is observed. This corresponds to an inflexion point for M(H), which separates the antiferroand (forced) ferromagnetic states. At lower tem-
1
:
I
I
:.
l+ ::.56.501 . . . . i .. : . . f : : : :.. : +’ .=.” f * + .'." l
; ++ = .+
.. . .: :: : :
5 :
::. . : ‘.
l
> N
: :
2
l .'0 E :: l .e + le f : 0
: : :. 0
: l
ZE
f
i t :
l*
‘.a . . . . ;* *. *.* : 0
l
co 0 13
LO1 13 . [l 1 13 l
0. -.
f
aI-
35.3
l.
I
.*
0
++‘. .o@e q..l*;.& 0.. **
l* l.
,
l
35.7
I
._
35.5
I
359 “/M ( k&/p,)
3k.7
Fig. 7. Arrott plots of isothermal magnetizations along the three main axes in the temperature The initial behaviour is isotropic in the paramagnetic phase, anisotropic in the ordered temperature for which the initial slope is infinite.
361
range of the NCel transition in DyAg. phase. The Ntel temperature is the
P. Morin et al. / Interplay of interactions in DyAg
applied field. This proves they are not driven by inhomogeneities in the sample, but constitute intrinsic properties. In low field the zero value of dM/dT, which indicates the maximum of M(T), provides us with the classical definition of TN: for the three directions, the TN value deduced from the 3 kOe curve is in full agreement with the previous determinations from the M(H) set. In high magnetic fields, it is worth noting that the temperature at which dM/dT vanishes corresponds to the crossing of isothermal M(H) curves. Thus this behaviour does not reveal any change of phase. In low fields, the existence of a smooth
peratures, the anomalies associated with T,* and T2* transitions are broad, due to the fact that H-scans are mainly parallel to the phase transition line in the (H, T) phase diagrams. In order to determine more accurately these phase diagrams, isofield magnetization curves, M(T), have been also collected along the three main crystallographic axes in fields around 3, 10, 20, 40, 50 and 60 kOe. They have been analyzed through their temperature derivative (fig. 9): the two first-order transitions at T,* and T2* are now well defined and their temperature difference, in particular along the [OOl] axis, increases with the
8 i
t t: t; : . ,: .. :
253
DyA9 coo13
.7
H = 56.7 kOe
..
.
4 I
0
-0
.. ; .’ !. . *..: t “GO...... $.:.....
l
0
H = 2.81 kOe l. :... . . . . *..
T949.1 K
T;= 47.4K
Fig. 9. Temperature derivative of the [OOl] isofield magnetization. The T,* and T;C lines separate the triple-q structure, the double-q one and the modulated arrangement from each other. The zero value of d M/dT defines at low field the N&l temperature.
254
P. Morin et al. / Interplay
40
50
45
of interactions
T (HI
in DyAg
55
II
50
55
Fig. 10. DyAg magnetic phase diagrams along the three main axes in fields lower than 70 kOe. Phase I is the triple-q structure, phase II the double-q one; phase III is modulated and phase IV paramagnetic. The 7-t*, T;” lines depend on the field orientation; open circles are deduced from M( If) curves, black dots and squares from M(T) ones.
q structure, phase II is still undetermined, phase III is modulated [5] and phase IV is paramagnetic. These two latter ones are demarcated by the line of inflexion points observed for M(H). Due to smooth variations, the analysis of the magnetization processes is rather complex close to the sec-
maximum for dM/dT is only the signature of a magnetization frozen in by the anisotropy up to temperatures close to TN. The low-field parts of the (H,T) phase diagrams are drawn in fig. 10. The T,*and T2*lines reveal anisotropic behaviours. Phase I is the triple-
Ii
IkOe)
Fig. 11. [Ool] magnetization curves at different temperatures in fields up to 200 kOe for DyAg.
255
P. Morin et al. / Interplay of interactions in DyAg
ond-order modulated-paramagnetic phase transition as previously indicated for the elastic constants. 3.3.3. Magnetic phase diagrams in high fields Isothermal magnetization curves have been obtained using the facilities of the S.N.C.I. in Grenoble for fields up to 20 T and also the facilities of the High Magnetic Field Laboratory of Osaka University for fields up to 40 T. Measurements at the S.N.C.I. were performed on the same monocrystalline sphere used at the Louis NCel Laboratory. For measurements in the (20 ms) long pulse magnet at Osaka, small sheets of about 0.32 mm in thickness, electrically insulated from each other, were assembled in order to form a cylindrical sample of 2.4 mm in diameter and 5 mm in length for the three main axes. Typical measurements are presented in figs. 11 and 12. At 4.2 K, the magnetizations measured in 40 T are 9.3, 9.0 and 6.4~~ along [ill], [loll and [OOl], respectively. The [ill] magnetic moment is smaller than the free ion value due to the residual 4f-wave function mixing by the CEF and also to the band polarization which is opposite to the 4f magnetization. The anisotropy of the magnetizaand tion is about 0.3pa, i.e. 3%, between [ill] I”
6-
T(K)
Fig. 13. Magnetic phase diagrams along the three main directions of DyAg. The different magnetic structures, the stability ranges of which are defined by the H lines, are drawn in fig. 14. The low-field part of the diagrams are given in fig. 10.
[loll. The alignment of magnetic moments along the [OOl] direction of the field is far from being achieved. Indeed at 4.2 K in 40 T, the ratio of [OOl] and [ill] magnetizations corresponds to an angle of 46.2” between the field and the moments still close of the (111) easy magnetization directions. This indicates how large the CEF anisotropy is. At high temperatures, the lack of accuracy due to the smoothness of the transitions prevents a precise determination of the magnetic phase diagrams. However, an interesting situation may be the nature of the intersections of the various line especially along the [OOl] and [loll axes between 40 and 50 K and 6 and 12 T. More accurate experiments are planned at the SNCI.
4. Discussion
H(kOe)
Fig. 12. Magnetization curves at 4.2 K along the three main directions in DyAg in fields up to 400 kOe. From the ratio of Too’] and M~[“‘] = 9.3p,, the magnetic moment appear to lie close to (221), then (111) directions, before the monotonous closing of the &spin umbrella.
The study of the magnetoelastic coupling has shown that it is large enough to induce a sizeable tetragonal strain in the case of a magnetic structure of tetragonal symmetry. Thus, the lack of any spontaneous strain unambiguously proves the magnetic structure to the triple-q with pairs of magnetic moments pointing along each of the 4 threefold axes (see fig. 14). The analysis of the quadrupolar properties also reveals that the pair interactions between quadru-
256
P. Morin et al. / Interplay of interactions
7 0 0 r
TmGa31
DYCU
80 a
_--__-----_-
300
5
350
400___
’
2
H/O TmGa4d H (kCk1
/” 19.5
Fig. 14. Critical fields along the three main directions at 4.2 K in DyAg and DyCu and at 1.5 K in TmGa,. The semi-dotted lines, corresponding to the H,, Hz, H3, H4 critical fields, separate the stability range of the drawn structures; F defines the forced ferromagnetic phase; the triple-q spontaneous structure is drawn in the low-field part along [loll. The structures are deduced from the magnetization processes in DyAg and DyCu and determined by neutron diffraction on a TmGa, single crystal. The magnetic field axes are hatched for values larger than the experimental limits. The TmGa, critical fields are scaled to the DyAg and DyCu ones, using the Neel temperature ratio (TN = 56.5 K in DyAg and 4.21 K in TmGas). Note that along [OOl], 120 kOe in TmGa, correspond to about 1600 kOe in DyAg.
poles are negative for the trigonal symmetry as usually observed in C&l-type structure compounds, but also for the tetragonal symmetry that constitutes a new feature for these intermetallics. As the CEF strongly favors the threefold axes, the triple-q structure is driven by the antiferroquadrupolar (AFQ) interactions of trigonal symmetry, which select a different [ill] easy axis among the (111) star from one site to the other. This multi-q arrangement is characteristic of CsCl-type anti-
in DyAg
ferromagnets as soon as the CEF determines (111) or (110) axes as easy magnetization directions as it was observed for instance in DyCu [lo], NdZn [21] and also in AuCu,-type compounds such TmGa, [ll]. This triple-q phase (phase I) is separated at high temperatures from a new phase, undetected up to now, by a first-order transition. This phase II is intrinsic to DyAg. Another difference between DyAg and DyCu is the existence in DyAg of the modulated magnetic phase in a small range below TN. This is a general trend in RAg and RCu compounds, which seems to be a little more pronounced in RAg. The transition between phase II and this modulated phase III is well defined and of first-order. The determination of the exact value for the NCel temperature appears rather complex in such as case of modulated-paramagnetic transition, this is also observed for the transport properties and will be discussed in the near future. The process of breaking the spontaneous tripleq structure by applying a magnetic field along the main crystallographic directions presents a behaviour very similar for DyAg, DyCu and TmGa 3. An analysis of DyCu and DyAg antiferromagnets in parallel with DyZn and DyCd ferromagnets was presented in ref. [lo] according to the nature of the CEF and of bilinear and quadrupolar pair interactions. The magnetic phase diagrams of DyCu (TN = 62.8 K) were also given in fields lower than 20 T. For both Dy antiferromagnets, the similar TN values and CEF level schemes allow us a direct comparison at least at low temperature (see fig. 10 of ref. [lo] and here fig. 14). The H,[“” and Hi1o’1 and Hf”‘] lines are identical. The other H[llll lines observed in DyAg are outside of the experimental range for DyCu. Along [OOl], the field range of the intermediate structure is larger in DyCu than in DyAg (at 4.2 K, Hi”‘] = 60, HJool] = 160 kOe and Hitoo’]= 76, Hjooll = 100 kOe, respectively). In both compounds, no other transition or sizeable rotation of the moment from [ill] axes are detected, due to the large CEF anisotropy in favor of threefold axes. Analyzing the magnetization processes, in particular the magnetization jumps at the critical fields led to the sequence of the intermediate and
P. Morin et al. / Interplay of interactions in DyAg
final structures in DyCu. That was achieved by considering the eight magnetic moments of the (2~2, 2a, 2~) cell in symmetrical arrangements with regard of the applied field. To simplify the hypotheses, the modulus of the moment was kept constant and only the (ii 0) propagation vectors were considered. The main result of the analysis was that the intermediate structures along the 3 main axes were generated by inverting pairs of spins along their own threefold axis to another. The CEF energy was unmodified in all the cases, the quadrupolar energy does not change (in the case of spin inversion) or is reduced (in the case of change of (111) axes); the main variation comes from the bilinear interactions between the two successive structures [lo]. The same analysis appears to be valid for DyAg because the values of the bilinear couplings and, above all, of both the critical fields and the corresponding magnetization jumps are very close in the two compounds. It is then possible to propose a magnetic structure for phase II. Indeed for H//[OOl] the intermediate structure (fig. 11 in ref. [lo]) is assumed to be in DyCu the superimposition of a ferromagnetic structure (qF = [000], M,//H//[OOl]) and a double-q antiferromagnetic arrangement (ql = [i 0 i], M,//[OlO] and q2 = [0 : i], M,//[lOO]). This intermediate structure disappears around 40 K in DyCu, whereas in DyAg it survives at high temperature down to zero field, where it reduces to the double-q structure by vanishing of the Mr component; in this phase II, the moments point along twofold axes, because the [loll and [ill] CEF energies are calculated to be very close and because negative quadrupolar (i.e. AFQ) interactions of tetragonal symmetry work for non-collinear arrangements along (110) axes in the same sense as the trigonal symmetry AFQ ones. Neutron diffraction experiments on a single crystal will allow us to verify this point in the future. In spite of very different bilinear interactions, TmGa, exhibits similar behaviours. For this antiferromagnet, the CEF favors threefold axes as easy magnetization directions. From the study of ErGa s, the bilinear interactions in RGa 3 are characterized by propagation vectors (i - 7, i, 0) which lead to the same modulated structure as
251
phase III in DyAg [22]. In TmGa,, AFQ interactions are large enough to stabilize the triple-q structure through a first-order transition at TN = 4.21 K. The weak pair interactions allowed us to fully construct the magnetic phase diagrams and to determine all the existing structures by neutron diffraction along the three main axes [ll]. The TmGa, set of magnetic phase diagrams is very similar to that of the two dysprosium compounds. This is underlined by comparing for the three alloys the tilt angle between the moments and the applied field in figs. l-3 of ref. [ll] and in fig. 12 for DyAg. This is reinforced by comparing the critical field values of TmGa3 to the DyCu and DyAg ones after scaling the field values according to the ratio of the NCel temperatures (fig. 14). All the structures determined in TmGa, confirmed the ones proposed from the analysis of magnetization processes in DyAg and DyCu. They are all built with the 8 spins of the magnetic cell identical with respect of the field except for the [loll intermediate structure (H > Hino’l). This latter one is composed of 2 different sets of spins: 4 are in a binary plane perpendicular to the field, the other 4 in the binary plane parallel to it. However, they were supposed to be parallel to threefold axes in Dy compounds and they point along twofold ones in TmGa,. The discrepancy is small because the CEF anisotropy between these two directions is calculated in the three compounds to be weak. Another difference is the occurrence for Dy ions of additional jumps in the intermediate phase along [ill] and [ 1011. They are probably due to anisotropy accidents of mainly CEF origin in the slow closing of the intermediate structure towards the field direction. Except for these pecularities, the process of breaking of the triple-q structure is then characterized by the same behaviour in the three compounds. The occurrence of multi-q structures is a rather common feature in several series of cubic intermetallic compounds as soon as a few conditions are realized: i) antiferromagnetic interactions are the basic one, ii) CEF parameters favouring threefold axes as easy magnetization directions are then needed, iii) negative quadrupolar interactions of trigonal symmetry then drive different orientations from one site to the other for the z-axis. This
258
P. Morin et al. / Interplay of interactions
is realized in the present three compounds and likely in other AuCu,-type and CsCl-type compounds. NdZn for instance fits the above conditions. Indeed it exhibits a change of easy magnetization axis at 18 K between a double-q structure with (110) easy axes at low temperature and a triple-q structure with (111) easy axes at high temperature [21]. Similar features exist in other series such as rare earth hexaborides. In Pr$, the low-temperature magnetic structure has been determined by neutron diffraction on a single crystal to be double-q ([$ a $1 and [a - $ i]) [23]. This arrangement is identical to the one, which occurs in the AFQ phase of Ce$ [24]. Within all these cubic series, AFQ interactions involve G’, the coefficient for the trigonal symmetry. Multi-q structures may also be stabilized by AFQ interactions of tetragonal symmetry, with moments pointing along different fourfold or twofold axes. The only evidence for such a case seems to be provided by the NaCl-type compounds HOP [25]. Below a first-order transition at 4.8 K in the ordered range (T, = 5.4 K), the flopside state is characterized by alternate ferromagnetic (111) planes. From one plane to the other, the moments jump from one fourfold axis to another. Kim and Levy have explained the structure and the magnetic phase diagram by considering positive bilinear interactions and negative tetragonal quadrupolar interactions [26]. In the same way, we plan to describe the magnetic phase diagrams in DyAg, DyCu and TmGa, using a 4-sublattice model which is needed for considering the 4 pairs of moments of the magnetic cell. From an experimental point of view, neutron diffraction will allow us to fully determine the magnetic phase diagrams of DyAg at least in the low field range.
References [l] P. Morin and D. Schmitt, J. Magn. Magn. Mat. 28 (1982) 188.
in DyAg
P. Morin, D. Schmitt and C. Vettier, J. de Phys. 46 (1985) 39. 121N. Nereson, J. Appl. Phys. 44 (1973) 4727. I31 N. Nereson, A.I.P. Conf. Proc. 10 (1973) 669. I41 P. Morin and D. Schmitt, J. Magn. Magn. Mat. 21 (1980) 243. [51 T. Kaneko, H. Yoshida, M. Ohashi and S. Abe, J. Magn. Magn. Mat. 70 (1987) 277. [61 G. Arnold, N. Nereson and C. Olsen, J. Chem. Phys. 46 (1967) 4041. and R. Chamard-Bois, Acta Cryst. A28 t71 M. Wintemberger (1972) 341. R. Chamard-Bois, M. Belakhovsky and PI M. Wintemberger, J. Pierre, Phys. Stat. Sol. 48 (1971) 705. I91 P. Morin and J. Pierre, Phys. Stat. Sol. (a) 21 (1974) 161. WI R. Aleonard, P. Morin and J. Rouchy, J. Magn. Magn. Mat. 46 (1984) 233. PII P. Morin, M. Giraud, P. Burlet and A. Czopnik, J. Magn. Magn. Mat. 68 (1987) 107. on Ferromagnetic WI P. Morin and D. Schmitt, Handbook Materials, vol. 5, eds. E.P. Wohlfarth and K.H.J. Buschow (North-Holland, Amsterdam, 1990) p. 1. [I31 K.R. Lea, M.R.J. Leask and W.P. Wolf, J. Phys. Chem. Solids 23 (1962) 1381. P41 P. Morin and D. Schmitt, Phys. Rev. B 23 (1981) 5936. (151 P. Morin, D. Schmitt and E. du Tremolet de Lacheisserie, Phys. Rev. B 21 (1980) 1742. Rev. Phys. Appl. 10 WI E. du Tremolet de Lacheisserie, (1975) 169. P71 P. Morin, J. Magn. Magn. Mat. 71 (1988) 151. WI C. Jaussaud, P. Morin and D. Schmitt, J. Magn. Magn. Mat. 22 (1980) 98. 1191 M. Giraud and P. Morin, J. Magn. Magn. Mat. 58 (1986) 135. m P. Morin and D. Schmitt, Phys. Rev. B 26 (1982) 3891. WI P. Morin and A. de Combarieu, Solid State Commun. 17 (1975) 975. WI P. Morin, M. Giraud, P.L. Regnault, E. Roudaut and A. Czopnik, J. Magn. Magn. Mat. 66 (1987) 345. S. Ktmii and v31 P. Burlet, J.M. Effantin, J. Rossat-Mignod, T. Kasuya, J. de Phys. 49 (1988) C8-59. P. Burlet, H. Bartholin, 1241J.M. Effantin, J. Rossat-Mignod, S. Kunii and T. Kasuya, J. Magn. Magn. Mat. 47848 (1985) 145. ~251P. Fischer, A. Furrer, E. Kaldis, J.K. Kjems and P.M. Levy, Phys. Rev. B 31 (1985) 456. WI D. Kim and P.M. Levy, J. Magn. Magn. Mat. 27 (1982) 257.
Materials
247
81 (1989) 247-258
INTERPLAY OF ANTIFERROMAGNETIC AND ANTIFERROQUADRUPOLAR INTERACTIONS IN DyAg AND OTHER RARE EARTH INTERMETALLIC COMPOUNDS P. MORIN,
J. ROUCHY,
K. YONENOBU
a, A. YAMAGISHI
a and M. DATE
a
Laboratoire Louis Nt!el ‘, CNRS, 166 X, 38042 Grenoble Cedex, France ’ Research Center for Extreme Materials, Osaka University, Toyonaka, Osaka 560, Japan
Received
25 May 1989; in revised form 11 July 1989
The study of the magnetoelasticity of the cubic (CsCl-type) rare earth intermetallic DyAg allows us to determine the strength of both the magnetoelastic coupling and the quadrupolar pair interactions. These latter ones are observed to be negative (antiferroquadrupolar type) for the tetragonal symmetry as well as for the trigonal one. They drive the magnetic structure to be triple-q at low temperature: the cubic magnetic cell consists of four pairs of ferromagnetic moments pointing along each of the treefold axes. At high temperature, it is replaced by a double-q structure, then, immediately below TN, by a modulated arrangement. The magnetization processes have been thoroughly studied along the three main axes in fields up to 40 T and compared with previous results in isomorphous DyCu and in the AuCu,-type compound TmGa,. The sequences under field of the different magnetic structures are identical and mainly determined by the crytalline electric field and the antiferroquadrupolar interactions. These 3 compounds do not set a peculiar case, but seem to belong to a larger family of cubic compounds with multiaxial structures governed by antiferroquadrupolar terms.
1. Introduction The intermetallic equiatomic rare earth-silver compounds crystallize within the C&l-type cubic structure. This simple arrangement and the nonmagnetic alloyed metal provide us with favourable conditions to thoroughly study the physical properties associated with the 4f shell. The RAg series is characterized by negative bilinear interactions. At least close to the NCel temperature, the actual magnetic structure is defined by a propagation vector which is not commensurate with the lattice cell. Indeed, modulated magnetic structures have been observed in TmAg [l], ErAg [2], HoAg [1,3]. The propagation vector is (i - 7, f, 0) with r around 0.07 and exhibits a weak temperature dependence [l]. In ErAg, a first-order transition at 9.5 K separates this modulated structure from the commensurate ($ f 0) one, stable at low temperature. Similar situations have been observed in isomorphous RCu compounds [4]. ’ Associated
with the Universite
Joseph
Fourier,
Grenoble.
0304-8853/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
In DyAg, two different ordered phases have been recently observed by both magnetic susceptibility and powder neutron diffraction measurements [5]. At low temperature, the (1 i 0) propagation vector previously determined by Arnold [6] is confirmed; between 46.5 K and TN = 56.6 K, a modulated structure is established in a way reminiscent of the ErAg situation. The presence of only two equivalent satellites for the (: : 0) position on the powder neutron diffraction pattern at 50 K (fig. 2c in ref. [5]) indicates that the propagation vector is (f - 7, i, 0) as in other RAg compounds. At low temperatures, the neutron diffraction pattern does not simply indicate that the magnetic structure is collinear. Indeed multiaxis (non-collinear) arrangements are possible, which are described by the presence of several propagation vectors from the (i t 0) star in each magnetic domain [7]. In the case of DyAg as for DyCu [4,8], the collinear (single-q) structure with magnetic moments along one fourfold axis is in competition with one double-q structure and one triple-q B.V.
248
P. Mm-in et al. / Interplay of interactions
structure (magnetic moments along two twofold axes and four threefold axes, respectively, as drawn in fig. 1 of ref. [4]). The lattice symmetry driven by the magnetic system through the magnetoelastic coupling is tetragonal in the case of the collinear and biaxial structures, but remains cubic in the case of the triple-q one. Experimentally, no lattice distortion has been detected in DyAg and DyCu by either neutron diffraction [5,6] or X-ray diffraction [9]. Our present aim is to show by means of parastriction and sound velocity measurements that the absence of any cell distortion is incompatible with the strength of the magnetoelastic coupling and only results from the cubic symmetry of the triple-q magnetic structure. We also want to establish that in spite of the intrinsic nature of the bilinear interactions which favours incommensurability, this triple-q structure is stabilized at low temperature by quadrupolar interactions, negative within the trigonal symmetry. Last, the study of the magnetic phase diagrams under a magnetic field will be presented and discussed in comparison with analogous diagrams obtained in other compounds such as DyCu [lo] and TmGa s [ll] which exhibit the same non-collinear spontaneous magnetic structure.
in DyAg
about 14 K. The other levels are at about 155 K. This level scheme favors a threefold axis as easy magnetization direction. The term: xo
= -&P&g_&J)
+ H)J
(3)
includes the Zeeman coupling in the external field H and the Heisenberg-type exchange written in the mean field approximation. From the exchange coefficient n = 0*/C determined by the temperature dependence of the magnetic susceptibility, a O* value of -27 K is deduced (section 3.3.1). For both tetragonal and trigonal symmetries, the quadrupolar terms have contributions from the magnetoelastic coupling with the lattice and from the pair interactions between rare earth sites: XQ = -((By)2/coy
+ IV)(@)@
-((Bf)2/c~+K~)((P,y)pIv+
+ 3(@)@) ...)?
(4)
BY, B’ and KY, K’ are the magnetoelastic and pair interaction coefficients, respectively. The magnetic properties in the paramagnetic state can be described using single ion susceptibilities, the expression of which are established from the cubic CEF level scheme using a perturbation theory. They allow us to determine the various quadrupolar coefficients by means of independent experimental ways [14,15].
2. Formalism The basic Hamiltonian describing the magnetic properties includes the following single-ion terms [12]:
3. Experiments
SP=.%&r
A magnetic field applied along a fourfold (or a trigonal) axis induces a change of length, which may be analyzed using the strain susceptibility x,(x,), the quadrupolar field-susceptibility xl” (xi”) and the first-order magnetic susceptibility, x0. For instance, for the tetragonal symmetry, the tetragonal strain ey may be analyzed with the following relation, which allows a linearized temperature variation in vanishing CEF effects [15].
+.%?o +xo.
(1)
The form .%cnr = wxo,
+ w( 1 - 1x I) 0,
(2)
describes the Crystalline Electric Field (CEF), using the parametrization of Lea Leask and Wolf [13]. As discussed in ref. [lo], the values W = - 0.48 K and x = -0.55 are coherent with the parameters determined in Ho-, Er- and TmAg isomorphous compounds and are also very close to the DyCu ones. These values will be kept constant in the following. They define a level scheme with the l?d3’ ground state and the I’$‘) first excited level at
3.1. Parastriction
H/p
= /m(l/@) x (1 - GYx$‘*(l
- nx,,),
(5)
249
P. Morin et al. / Interplay of interactions in DyAg
1 0
60
180 120 T(K)
240
I 300 *.
Fig. 1. Temperature dependence of H/@ for the tetragonal symmetry in DyAg. ry is the tetragonal strain. The experimental variation is described with the parameters indicated and using eq. (5) in the text.
3.6
I 0
...
I
100
200
-...
se
300
T(K)
Fig. 2. Temperature variation of the C’ = C, and C” = (C,, + 2C,s)/3 normal elastic modes in DyAg.
C,Y = Cfl - Cfz is the non-magnetic lattice elastic constant, GY = P/CJ + KY the total quadrupolar coefficient. Changes of length induced by magnetic fields up to 10 kOe have been measured along the [OOl] axis of a monocrystalline sphere of 4.7 mm in diameter in a capacitance dilatometer [16]. The data of fig. 1 are described using O* = - 27 K (see section 3.3.1) and /m = 115. With C,Y = 1.615 X lo5 K/at (see section 3.2) the magnetoelastic coefficient follows as BY = + 12 K/at. The fit is slightly improved by introducing weakly negative quadrupolar interactions, characterized by GY around - 10 mK. According to the experimental uncertainties and the interplay between the parameters, BY may range from 12 to 9 K whereas GY varies from - 10 to 0 mK. The total quadrupolar interactions of tetragonal symmetry are small, even slightly negative. As the CEF terms themselves, they do not favor a fourfold axis as the direction of easy magnetization and then forbid the collinear antiferromagnetic structure to occur. The BY value is reminiscent of the parastriction analysis in DyCu [lo] (B, = BY = 15 K) and in DyZn [15] (By = 10 K). The immediate consequence is that in the case of the simple-q magnetic structure or of the double-q one, the tetragonal spontaneous strain is calculated to reach 8 X lop3 (as in DyZn) or - 4 X 10e3, in full contradiction with the zero experimental value. Thus among the three magnetic structures compatible with the low temperatures
neutron diffraction spectrum, the triple-q structure is the actual one. Applying the magnetic field along [ill] leads to changes of length too small to be significantly analyzed. In DyAg as in DyCu and DyZn, the B’ magnetoelastic coupling is very weak. This also agrees with other determinations in isomorphous series [12]. 3.2. Elastic constants Samples with faces (111) planes have been ties corresponding to tions C,, - C12, C,, + and C,, + 2C,, + 4C,
I
parallel to (lOO), (110) and used. The ultrasonic velociC,,, C, and the combinaC12 + 2C44, C,, - C12 + C44 have been measured be-
I
1
u
1
J
t
1.6-
0
100
T (K)
200
300
Fig. 3. Temperature variation of the Cy = (C,, - Cl,)/2 elastic mode in DyAg. Data in the paramagnetic range are described using eq. (6) with the parameters indicated.
250
P. Morin et al. / Interplay of interactions in DyAg
tween 2 and 300 K. These led to the temperature variation of the three cubic normal modes Cy = C,, - C,,, C’ = C, and C” = )(C,, + 2C,,) (figs. 2 and 3). As previously discussed, the absolute values of the three cubic modes at room temperature are characteristics of the stability of the CsCl-type structure in RAg [1’7]. Indeed, it is well known that this arrangement becomes less stable when proceeding up the series and the instability is particularly obvious for La and CeAg. Considering the large values of their elastic constants, the lattices of DyAg and YAg - the present non-magnetic reference - can be estimated as quite stable. For DyAg as for DyCu, all the modes remain observable at low temperatures below TN. This indicates the absence of any sizeable magnetoelastic interaction with magnetic domains. This is in contrast to the case of magnetic compounds tetragonally strained such as TmCu [18], TmAg [19], where the ultrasonic waves, Cy in particular, lose all their energy in vibrating the domain walls. Thus, this ultrasonic technique constitutes quite a relevant probe to determine the existence of a
magnetic triple-q structure which maintains the cubic symmetry. Other fruitful information, which is provided in the ordered range, is the existence of two clear anomalies at T,*= 47.4 K and T,*= 49.0K below the N6el temperature in fig. 4. The lower temperature corresponds to the transition previously observed at 46.5 K [5]. The transition at T2* has not been detected up to now although it is clearly evidenced here. We have observed this pair of transitions in all the different single crystals and annealed polycrystals we have prepared; the analysis of magnetization measurements will show that both of them are intrinsic properties of the compound. The actual value of the Neel temperature, higher than 55 K, is difficult to accurately determine from the temperature variations in fig. 4. This is mainly due to the fact that the second order transition between incommensurate and paramagnetic phases is very smooth; this is associated with the presence of weak magnetic moments on given 4f sites. In the paramagnetic phase, the magnetoelastic contribution to the measured elastic constants is rather weak; the temperature dependence of the bulk modulus is quite normal. This is indicative of the weakness of the cubic magnetoelastic coupling and has been observed in all CsCl-type rare earth compounds [12]. The C’ mode exhibits a linear temperature variation down to 100 K. The small downwards curvature observed above TN cannot be described within the susceptibility formalism and seems to be due to precursory effects associated with magnetic ordering. The temperature dependence of Cy is not linear in contrast with the behaviour observed in YAg. The small softening induced by the magnetoelastic coupling may be described by the usual expression: cy = c,y - PXJ(l
Fig. 4. Temperature variation of the 3 normal elastic modes in the range of the magnetic transitions. The behaviours are very different for the 3 modes, in particular around T,* and 7”*, the temperatures demarcating the commensurate phases.
-KY&).
(6)
The calculated strain susceptibility xv exhibits a Van Vleck behaviour at low temperature, which prevents the magnetoelastic contribution from drastically varying as a function of temperature. Considering the anharmonicity of the lattice through a C,Y background with the same slope as
P. Morin et al. / Interplay of interactions in DyAg
251
in YAg leads to a magnetoelastic coefficient, BY = 5 K (fig. 3). Changing the background slope by a ratio of 2 gives a similar agreement with BY = 11 K. Note that negative KY values could slightly increase BY for a constant C,Y.This weakness of the softening forbids accurate determinations of the quadrupolar parameters although their values are in agreement with their determination by parastriction. 3.3. Magnetization Magnetization processes have been extensively studied for several ranges of the magnetic field and temperatures. First, numerous data have been collected in the Louis NCel Laboratory in fields up to 76 kOe applied along the three main crystallographic directions of a monocrystalline sphere. Through Arrott’s plots as well as field or temperature derivatives, they allow us to determine the strength of the bilinear interactions, the quadrupolar contribution to the third-order magnetic susceptibility and to obtain precise information on the magnetic phase diagrams, in particular around the N6el temperature. 3.3.1. Paramagnetic
phase
Isothermal magnetization curves have been collected between 60 and 100 K. From the extrapolated null magnetization value in Arrott’s plots, the reciprocal magnetic susceptibility was found perfectly isotropic and its temperature variation is drawn in fig. 5. The difference between this exI
60
I
4 Ag
60
70 1 (K)
80
90
Fig. 6. Temperature variation of the third-order magnetic susceptibilities for the trigonal and tetragonal symmetries in DyAg. The different lines are calculated with the parameters indicated according to eq. (7). In both cases, the quadrupolar contributions are negative.
perimental behaviour and the calculated one (l/x = l/x,, - 0*/C) leads to a value 8 = - 27 K, which characterizes the bilinear interactions. The analysis of the slope of Arrott’s plots as well as of M/H (Hz) curves leads to the temperature variation of the initial curvature of the magnetization curve, i.e., of the third-order magnetic susceptibility [14,20]. This latter one was shown to be sensitive to quadrupolar contributions, which modify its usual behaviour described in the CEF model by xl” and x,(3) for the tetragonal and trigonal symmetries, respectively:
xfj = [l - (o*/c)xo] -4 x
[xl”+24 ~3~/(1-G’XI)]
XQ = [l - (o*/c)x,] 301 50
70
90
I
T (K) Fig. 5. Temperature variation of the reciprocal first-order magnetic susceptibility in DyAg. The shift between data and the single-ion behaviour (hatched line) defines the bilinear interactions coefficient, 8 * = - 27 K.
7
/,\
-4
x [ xi3) + 6G’( ~l”)~/(l-
3~7x4.
The very small values, which are observed for both symmetries close to the limit of sensitivity, are due to the large negative bilinear interactions (@* = -27 K) (fig. 6). For the tetragonal symmetry, the
252
P. Morin et al. / Interplay of interactions
in DyAg
data indicate that the quadrupolar interactions are slightly negative. Below 70 K, a small shift between the data and calculated values is observed as previously seen for the temperature variations of the reciprocal first-order susceptibility. It exists also for trigonal symmetry and may be the signature of magnetic fluctuations. For [ill], the quadrupolar contribution is obviously negative, as systematically observed in CsCl-type rare earth intermetallics [12].
... :. : . : : : .. . .
.*...‘..
45.2OK
3.3.2. Magnetic phase diagrams in low fields The determination of the NCel temperature was attempted by means of Arrott’s plots of isothermal magnetization curves in fields applied along [OOl], [Oil] and [ill] (fig. 7). At high temperatures, Arrott plots are linear with a positive slope (negative curvature for M(H)). At low temperature, the slope is negative in the range where the magnetization is small (positive curvature for M(H)). The N6el temperature is defined by the infinite slope of the corresponding Arrott plot. Thus TN lies in the range (56.3, 56.5 K). This result was confirmed by analyzing also the field derivatives (fig. 8). At TN, dM/dH is con-
4
.... . . .. . . *
2.)
. .. ..
$
.
...’ ?.
.... 27*............i.........
‘..‘....‘...,.
.‘..
2.“
.,,...............*
. .. . .._
5?.BOK :_. /..:.* ......:. ....:. :’
t .* ..‘... i
t
25l 0
20
H( kOe)
I
60
40
Fig. 8. Field derivative of the [OOl] isothermal magnetization the temperature range of the NCel transition of DyAg.
in
stant in a large range. Below TN, a smooth maximum is observed. This corresponds to an inflexion point for M(H), which separates the antiferroand (forced) ferromagnetic states. At lower tem-
1
:
I
I
:.
l+ ::.56.501 . . . . i .. : . . f : : : :.. : +’ .=.” f * + .'." l
; ++ = .+
.. . .: :: : :
5 :
::. . : ‘.
l
> N
: :
2
l .'0 E :: l .e + le f : 0
: : :. 0
: l
ZE
f
i t :
l*
‘.a . . . . ;* *. *.* : 0
l
co 0 13
LO1 13 . [l 1 13 l
0. -.
f
aI-
35.3
l.
I
.*
0
++‘. .o@e q..l*;.& 0.. **
l* l.
,
l
35.7
I
._
35.5
I
359 “/M ( k&/p,)
3k.7
Fig. 7. Arrott plots of isothermal magnetizations along the three main axes in the temperature The initial behaviour is isotropic in the paramagnetic phase, anisotropic in the ordered temperature for which the initial slope is infinite.
361
range of the NCel transition in DyAg. phase. The Ntel temperature is the
P. Morin et al. / Interplay of interactions in DyAg
applied field. This proves they are not driven by inhomogeneities in the sample, but constitute intrinsic properties. In low field the zero value of dM/dT, which indicates the maximum of M(T), provides us with the classical definition of TN: for the three directions, the TN value deduced from the 3 kOe curve is in full agreement with the previous determinations from the M(H) set. In high magnetic fields, it is worth noting that the temperature at which dM/dT vanishes corresponds to the crossing of isothermal M(H) curves. Thus this behaviour does not reveal any change of phase. In low fields, the existence of a smooth
peratures, the anomalies associated with T,* and T2* transitions are broad, due to the fact that H-scans are mainly parallel to the phase transition line in the (H, T) phase diagrams. In order to determine more accurately these phase diagrams, isofield magnetization curves, M(T), have been also collected along the three main crystallographic axes in fields around 3, 10, 20, 40, 50 and 60 kOe. They have been analyzed through their temperature derivative (fig. 9): the two first-order transitions at T,* and T2* are now well defined and their temperature difference, in particular along the [OOl] axis, increases with the
8 i
t t: t; : . ,: .. :
253
DyA9 coo13
.7
H = 56.7 kOe
..
.
4 I
0
-0
.. ; .’ !. . *..: t “GO...... $.:.....
l
0
H = 2.81 kOe l. :... . . . . *..
T949.1 K
T;= 47.4K
Fig. 9. Temperature derivative of the [OOl] isofield magnetization. The T,* and T;C lines separate the triple-q structure, the double-q one and the modulated arrangement from each other. The zero value of d M/dT defines at low field the N&l temperature.
254
P. Morin et al. / Interplay
40
50
45
of interactions
T (HI
in DyAg
55
II
50
55
Fig. 10. DyAg magnetic phase diagrams along the three main axes in fields lower than 70 kOe. Phase I is the triple-q structure, phase II the double-q one; phase III is modulated and phase IV paramagnetic. The 7-t*, T;” lines depend on the field orientation; open circles are deduced from M( If) curves, black dots and squares from M(T) ones.
q structure, phase II is still undetermined, phase III is modulated [5] and phase IV is paramagnetic. These two latter ones are demarcated by the line of inflexion points observed for M(H). Due to smooth variations, the analysis of the magnetization processes is rather complex close to the sec-
maximum for dM/dT is only the signature of a magnetization frozen in by the anisotropy up to temperatures close to TN. The low-field parts of the (H,T) phase diagrams are drawn in fig. 10. The T,*and T2*lines reveal anisotropic behaviours. Phase I is the triple-
Ii
IkOe)
Fig. 11. [Ool] magnetization curves at different temperatures in fields up to 200 kOe for DyAg.
255
P. Morin et al. / Interplay of interactions in DyAg
ond-order modulated-paramagnetic phase transition as previously indicated for the elastic constants. 3.3.3. Magnetic phase diagrams in high fields Isothermal magnetization curves have been obtained using the facilities of the S.N.C.I. in Grenoble for fields up to 20 T and also the facilities of the High Magnetic Field Laboratory of Osaka University for fields up to 40 T. Measurements at the S.N.C.I. were performed on the same monocrystalline sphere used at the Louis NCel Laboratory. For measurements in the (20 ms) long pulse magnet at Osaka, small sheets of about 0.32 mm in thickness, electrically insulated from each other, were assembled in order to form a cylindrical sample of 2.4 mm in diameter and 5 mm in length for the three main axes. Typical measurements are presented in figs. 11 and 12. At 4.2 K, the magnetizations measured in 40 T are 9.3, 9.0 and 6.4~~ along [ill], [loll and [OOl], respectively. The [ill] magnetic moment is smaller than the free ion value due to the residual 4f-wave function mixing by the CEF and also to the band polarization which is opposite to the 4f magnetization. The anisotropy of the magnetizaand tion is about 0.3pa, i.e. 3%, between [ill] I”
6-
T(K)
Fig. 13. Magnetic phase diagrams along the three main directions of DyAg. The different magnetic structures, the stability ranges of which are defined by the H lines, are drawn in fig. 14. The low-field part of the diagrams are given in fig. 10.
[loll. The alignment of magnetic moments along the [OOl] direction of the field is far from being achieved. Indeed at 4.2 K in 40 T, the ratio of [OOl] and [ill] magnetizations corresponds to an angle of 46.2” between the field and the moments still close of the (111) easy magnetization directions. This indicates how large the CEF anisotropy is. At high temperatures, the lack of accuracy due to the smoothness of the transitions prevents a precise determination of the magnetic phase diagrams. However, an interesting situation may be the nature of the intersections of the various line especially along the [OOl] and [loll axes between 40 and 50 K and 6 and 12 T. More accurate experiments are planned at the SNCI.
4. Discussion
H(kOe)
Fig. 12. Magnetization curves at 4.2 K along the three main directions in DyAg in fields up to 400 kOe. From the ratio of Too’] and M~[“‘] = 9.3p,, the magnetic moment appear to lie close to (221), then (111) directions, before the monotonous closing of the &spin umbrella.
The study of the magnetoelastic coupling has shown that it is large enough to induce a sizeable tetragonal strain in the case of a magnetic structure of tetragonal symmetry. Thus, the lack of any spontaneous strain unambiguously proves the magnetic structure to the triple-q with pairs of magnetic moments pointing along each of the 4 threefold axes (see fig. 14). The analysis of the quadrupolar properties also reveals that the pair interactions between quadru-
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P. Morin et al. / Interplay of interactions
7 0 0 r
TmGa31
DYCU
80 a
_--__-----_-
300
5
350
400___
’
2
H/O TmGa4d H (kCk1
/” 19.5
Fig. 14. Critical fields along the three main directions at 4.2 K in DyAg and DyCu and at 1.5 K in TmGa,. The semi-dotted lines, corresponding to the H,, Hz, H3, H4 critical fields, separate the stability range of the drawn structures; F defines the forced ferromagnetic phase; the triple-q spontaneous structure is drawn in the low-field part along [loll. The structures are deduced from the magnetization processes in DyAg and DyCu and determined by neutron diffraction on a TmGa, single crystal. The magnetic field axes are hatched for values larger than the experimental limits. The TmGa, critical fields are scaled to the DyAg and DyCu ones, using the Neel temperature ratio (TN = 56.5 K in DyAg and 4.21 K in TmGas). Note that along [OOl], 120 kOe in TmGa, correspond to about 1600 kOe in DyAg.
poles are negative for the trigonal symmetry as usually observed in C&l-type structure compounds, but also for the tetragonal symmetry that constitutes a new feature for these intermetallics. As the CEF strongly favors the threefold axes, the triple-q structure is driven by the antiferroquadrupolar (AFQ) interactions of trigonal symmetry, which select a different [ill] easy axis among the (111) star from one site to the other. This multi-q arrangement is characteristic of CsCl-type anti-
in DyAg
ferromagnets as soon as the CEF determines (111) or (110) axes as easy magnetization directions as it was observed for instance in DyCu [lo], NdZn [21] and also in AuCu,-type compounds such TmGa, [ll]. This triple-q phase (phase I) is separated at high temperatures from a new phase, undetected up to now, by a first-order transition. This phase II is intrinsic to DyAg. Another difference between DyAg and DyCu is the existence in DyAg of the modulated magnetic phase in a small range below TN. This is a general trend in RAg and RCu compounds, which seems to be a little more pronounced in RAg. The transition between phase II and this modulated phase III is well defined and of first-order. The determination of the exact value for the NCel temperature appears rather complex in such as case of modulated-paramagnetic transition, this is also observed for the transport properties and will be discussed in the near future. The process of breaking the spontaneous tripleq structure by applying a magnetic field along the main crystallographic directions presents a behaviour very similar for DyAg, DyCu and TmGa 3. An analysis of DyCu and DyAg antiferromagnets in parallel with DyZn and DyCd ferromagnets was presented in ref. [lo] according to the nature of the CEF and of bilinear and quadrupolar pair interactions. The magnetic phase diagrams of DyCu (TN = 62.8 K) were also given in fields lower than 20 T. For both Dy antiferromagnets, the similar TN values and CEF level schemes allow us a direct comparison at least at low temperature (see fig. 10 of ref. [lo] and here fig. 14). The H,[“” and Hi1o’1 and Hf”‘] lines are identical. The other H[llll lines observed in DyAg are outside of the experimental range for DyCu. Along [OOl], the field range of the intermediate structure is larger in DyCu than in DyAg (at 4.2 K, Hi”‘] = 60, HJool] = 160 kOe and Hitoo’]= 76, Hjooll = 100 kOe, respectively). In both compounds, no other transition or sizeable rotation of the moment from [ill] axes are detected, due to the large CEF anisotropy in favor of threefold axes. Analyzing the magnetization processes, in particular the magnetization jumps at the critical fields led to the sequence of the intermediate and
P. Morin et al. / Interplay of interactions in DyAg
final structures in DyCu. That was achieved by considering the eight magnetic moments of the (2~2, 2a, 2~) cell in symmetrical arrangements with regard of the applied field. To simplify the hypotheses, the modulus of the moment was kept constant and only the (ii 0) propagation vectors were considered. The main result of the analysis was that the intermediate structures along the 3 main axes were generated by inverting pairs of spins along their own threefold axis to another. The CEF energy was unmodified in all the cases, the quadrupolar energy does not change (in the case of spin inversion) or is reduced (in the case of change of (111) axes); the main variation comes from the bilinear interactions between the two successive structures [lo]. The same analysis appears to be valid for DyAg because the values of the bilinear couplings and, above all, of both the critical fields and the corresponding magnetization jumps are very close in the two compounds. It is then possible to propose a magnetic structure for phase II. Indeed for H//[OOl] the intermediate structure (fig. 11 in ref. [lo]) is assumed to be in DyCu the superimposition of a ferromagnetic structure (qF = [000], M,//H//[OOl]) and a double-q antiferromagnetic arrangement (ql = [i 0 i], M,//[OlO] and q2 = [0 : i], M,//[lOO]). This intermediate structure disappears around 40 K in DyCu, whereas in DyAg it survives at high temperature down to zero field, where it reduces to the double-q structure by vanishing of the Mr component; in this phase II, the moments point along twofold axes, because the [loll and [ill] CEF energies are calculated to be very close and because negative quadrupolar (i.e. AFQ) interactions of tetragonal symmetry work for non-collinear arrangements along (110) axes in the same sense as the trigonal symmetry AFQ ones. Neutron diffraction experiments on a single crystal will allow us to verify this point in the future. In spite of very different bilinear interactions, TmGa, exhibits similar behaviours. For this antiferromagnet, the CEF favors threefold axes as easy magnetization directions. From the study of ErGa s, the bilinear interactions in RGa 3 are characterized by propagation vectors (i - 7, i, 0) which lead to the same modulated structure as
251
phase III in DyAg [22]. In TmGa,, AFQ interactions are large enough to stabilize the triple-q structure through a first-order transition at TN = 4.21 K. The weak pair interactions allowed us to fully construct the magnetic phase diagrams and to determine all the existing structures by neutron diffraction along the three main axes [ll]. The TmGa, set of magnetic phase diagrams is very similar to that of the two dysprosium compounds. This is underlined by comparing for the three alloys the tilt angle between the moments and the applied field in figs. l-3 of ref. [ll] and in fig. 12 for DyAg. This is reinforced by comparing the critical field values of TmGa3 to the DyCu and DyAg ones after scaling the field values according to the ratio of the NCel temperatures (fig. 14). All the structures determined in TmGa, confirmed the ones proposed from the analysis of magnetization processes in DyAg and DyCu. They are all built with the 8 spins of the magnetic cell identical with respect of the field except for the [loll intermediate structure (H > Hino’l). This latter one is composed of 2 different sets of spins: 4 are in a binary plane perpendicular to the field, the other 4 in the binary plane parallel to it. However, they were supposed to be parallel to threefold axes in Dy compounds and they point along twofold ones in TmGa,. The discrepancy is small because the CEF anisotropy between these two directions is calculated in the three compounds to be weak. Another difference is the occurrence for Dy ions of additional jumps in the intermediate phase along [ill] and [ 1011. They are probably due to anisotropy accidents of mainly CEF origin in the slow closing of the intermediate structure towards the field direction. Except for these pecularities, the process of breaking of the triple-q structure is then characterized by the same behaviour in the three compounds. The occurrence of multi-q structures is a rather common feature in several series of cubic intermetallic compounds as soon as a few conditions are realized: i) antiferromagnetic interactions are the basic one, ii) CEF parameters favouring threefold axes as easy magnetization directions are then needed, iii) negative quadrupolar interactions of trigonal symmetry then drive different orientations from one site to the other for the z-axis. This
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is realized in the present three compounds and likely in other AuCu,-type and CsCl-type compounds. NdZn for instance fits the above conditions. Indeed it exhibits a change of easy magnetization axis at 18 K between a double-q structure with (110) easy axes at low temperature and a triple-q structure with (111) easy axes at high temperature [21]. Similar features exist in other series such as rare earth hexaborides. In Pr$, the low-temperature magnetic structure has been determined by neutron diffraction on a single crystal to be double-q ([$ a $1 and [a - $ i]) [23]. This arrangement is identical to the one, which occurs in the AFQ phase of Ce$ [24]. Within all these cubic series, AFQ interactions involve G’, the coefficient for the trigonal symmetry. Multi-q structures may also be stabilized by AFQ interactions of tetragonal symmetry, with moments pointing along different fourfold or twofold axes. The only evidence for such a case seems to be provided by the NaCl-type compounds HOP [25]. Below a first-order transition at 4.8 K in the ordered range (T, = 5.4 K), the flopside state is characterized by alternate ferromagnetic (111) planes. From one plane to the other, the moments jump from one fourfold axis to another. Kim and Levy have explained the structure and the magnetic phase diagram by considering positive bilinear interactions and negative tetragonal quadrupolar interactions [26]. In the same way, we plan to describe the magnetic phase diagrams in DyAg, DyCu and TmGa, using a 4-sublattice model which is needed for considering the 4 pairs of moments of the magnetic cell. From an experimental point of view, neutron diffraction will allow us to fully determine the magnetic phase diagrams of DyAg at least in the low field range.
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in DyAg
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