Antiferromagnetic ordering between unstable 4f shells in CeAl2

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Antiferromagnetic ordering between unstable 4f shells in CeAl2

F. Steglich, C. Bredl, M. Loewenhaupt, K. Schotte

To cite this version:

F. Steglich, C. Bredl, M. Loewenhaupt, K. Schotte. Antiferromagnetic ordering between un- stable 4f shells in CeAl2. Journal de Physique Colloques, 1979, 40 (C5), pp.C5-301-C5-307.

�10.1051/jphyscol:19795106�. �jpa-00218889�

JOURNAL DE PHYSIQUE Colloque C5, supplément au n° 5, Tome 40, Mai 1979, page C5-301

Antiferromagnetic ordering between unstable 4f shells in CeAl

²

0)

F. Steglich (*), C. D. Bredl (*), M. Loewenhaupt (**) and K. D. Schotte (***)

(*) II. Physikahsches Institut, Universitat zu Koln, F.R.G.

(**) Institut fur Festkorperforschung, KFA Julich, F.R.G.

(***) Institut fur Theoretische Physik, Freie Universitat Berlin, F.R.G.

Résumé. — La situation expérimentale du composé CeAl2 est décrite principalement d'après des expériences de diffusion inélastique des neutrons et de chaleur spécifique. L'aspect le plus intéressant est le comportement du CeAl2 à basses températures. Au-dessous de 3,9 K, une structure antiferromagnétique coexiste avec une inter- action dynamique entre les moments magnétiques de cérium et les électrons libres. Nous maintenons que cette action dynamique est causée par l'effet Kondo et non par des fluctuations de valence. La situation expérimentale peut être décrite à l'aide d'un modèle théorique de moment local.

Abstract. — We review the experimental situation of the unusual compound CeAl2 based on inelastic neutron scattering and specific heat measurements. The most interesting aspect of CeAl2 is its low temperature behavior.

Below 3.9 K, a complicated antiferromagnetic structure coexists with a dynamic interaction between Ce moments and conduction electrons. It is argued that this dynamic interaction is due to the Kondo effect rather than to valence fluctuations. The experimental situation can be phenomenologically well described with the aid of a local model.

1. Introduction. — Although the cubic Laves phase Kondo alloy. For example, coexistence between CeAl

2

has been extensively studied in the past, its superconductivity and the Kondo effect was first quite unusual behavior is up to now not completely established with this system [5]. The theoretical understood. Especially in its low temperature phase, implications of the Kondo compounds have been CeAl

²

shows strange properties : below 3.9 K, a discussed by Jullien et al. [6].

sinusoidal incommensurable, antiferromagnetic struc- In order to understand the thermodynamics of ture was observed by neutron diffraction experiments CeAl

²

qualitatively it appears to be sufficient to rely of Barbara et al. [1]. By this finding the inelastic on the molecular field approximation [3]. The mole- peak earlier discovered by neutron scattering [2] cular field acts on the single Ce ions whose properties was confirmed to be a magnon line. In a recent study are described by the wave function in their T

7

crystal of the specific heat down to T = 0.3 K and in magnetic field (CF) ground state doublet. The T

1

state is fields up to B = 5 tesla, a large contribution linear broadened by the dynamic interaction with the in T and almost independent of B was found [3]. Fermi sea measured by T

K, the single ion Kondo

This hints at a dynamic coupling of the Ce moments temperature.

to the conduction electrons which apparently coexists In this paper, we review the experimental status with the long range magnetic order between Ce ofCeAl

2

with special emphasis of its novel low tempe- moments (the latter being also mediated by the rature phase (section 4). An estimate of the single conduction electrons, namely via their RKKY pola- ion T

^K

value of CeAl

²

is given in section 2, and some rization). new observations concerning its paramagnetic phase Recently, Probst and Wittig [4] have shown that are presented in section 3. A collection of open the magnetic structure visible in the temperature questions will be found in section 5. They mark this dependence of the resistivity of CeAl

²

can be signifi- article as an intermediate report on one interesting cantly influenced, and finally suppressed, only at an aspect of the more general field of unstable RE ions external pressure Z 30 kbar. This seems unequivo- in metals. To provide a comprehensive information, cally to rule out the intermediate valence (IV) pheno- despite space limitation, we restrict ourselves to two menon as origin of the large electronic specific heat complementary experimental techniques, a micro- in CeAl

²

. On the other hand, dilute solutions of scopic one (inelastic neutron scattering), and a thermo- CeAl

²

in the isostructural, non-magnetic LaAl

²

pre- dynamic one (specific heat).

sent the best known example for a rare earth (RE)

2. Estimate of the single ion T

^K

value in CeAl

²

. — (') Work supported by SFB 125 Aachen/Julich/Kdln. The single ion T

^K

value cannot be obtained from

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19795106

C5-302 F. STEGLICH, C. D. BREDL, M. LOEWENHAUPT AND K. D. SCHOTTE

measurements in the ordered state directly. One can use neutron scattering data taken above the Ntel temperature. The half width r / 2 of the quasielastic neutron line can be used to provide an upper limit of TK. A second way to estimate T, simply consists in diluting the magnet. Finally, one can get rid of interaction effects by applying a sufficiently strong magnetic field. This allows to deduce T, from specific heat data.

In figure 1, the half width of the quasielastic neutron line is shown as a function of temperature.

For convenience, the straight line r / 2

=

k , T is also shown. Both lines cross near 5 K which means that at lower temperatures the fluctuations of the Ce moments are faster than the thermal fluctuations.

Disregarding interaction effects, the residual (T

+

0) line width r/2(0) gives an upper limit of the single ion TK, namely T, < 6 K. Although it was pre- viously found that the major part of r/2(0) is caused by the single ion effects [7], there may also be some measurable contribution from Ce-Ce correlations (in the form of diffusive modes) which will however not exceed 30 % [S].

This TK of the CeAl, compound is one order of magnitude larger than that of the well established dilute Kondo alloy (La, Ce)Al, [9]. Assuming the band structures of c e Z 2 and LaAl, to be similar, the easiest explanation of the TK increase is an increase of the exchange integral I ^J,, I which is connected to TK via

(Y, Ce)AI,. For a dilute (La, Y, Ce)Al, sample exhi- b3ing the lattice spacing o f e A I , , TK

= (5

+ ²⁾ K was found which is comparable to r/2(0) of the CeAl, compound.

Another estimate for the sihgle ion T, of CeA1, was obtained from the analysis of high field specific heats in (La,-,Ce,)Al, [3]. These results display the interaction of Ce with both, other Ce ions and the conduction electrons, namely by a cusp and a hump (which occurs at elevated temperature). The cusp is observed for Ce concentrations large and magnetic fields small enough. The hump is seen at low Ce concentration and in high magnetic field. Figure 2 shows how the hump emerges and simultaneously the cusp is suppressed upon application of increasing external fields.

where W is of the order of the band width and N the conduction band density of states at the Fermi level. The increase of I J,, I is then assumed to be caused by a lattice pressure upon increasing Ce concentration (or decreasing average lattice spacing).

The dominating influence of the average lattice spacing on TK was demonstrated by resistivity measu- rements with dilute (La, Y, Ce)Al, alloys [l01

:

TK increases from 0.4 R T L a , - Ce)Al, to 100 K in

Fig. 1. - Quasielastic line width vs. temperature for CeAI,.

Fig. 2. - Contribution of one Ce ion to the specific heat of a (La, ,Ce,.,)AI, single crystal, C*, in units of k,, as a function of temperature and magnetic field (in tesla) applied parallel to the [l001 direction.

It was found that the cusp temperature T,

depends quadratically on the Ce concentration in

(La, -,Ce,)Al, [3]. This had been interpreted by Ce

pairs which are coupled even over quite a long distance

by the same spin density wave (SDW) which had been

directly observed in CeAl, [l]. Within this type of

reasoning the cusp indicates the onset of long range

(and complex) antiferromagnetism even in the less

ANTIFERROMAGNETIC ORDERING BETWEEN UNSTABLE 4f SHELLS IN CeAI, C5-303

concentrated alloys. According to the susceptibi- lity [l l], this ordering does occur even for a Ce concen- tration as small as 10 at %.

In a high magnetic field, antiferromagnetism is destroyed and the magnetic moments behave as in a paramagnet, i.e. are aligned by the field. The high field humps of the specific heat have the shape of broadened Schottky anomalies. This broadening shows that the Zeeman levels only have a finite life- time, namely by the coupling to the conduction electrons. Assuming that this broadening of Zeeman levels is of Lorentzian shape and its width (HWHM) of the order of T,, the broadened Schottky anomalies can be analyzed on the basis of a resonance level model [12], which was originally developed for dilute Kondo alloys. We mention the three salient results of this analysis [3]

:

1) The entropy per Ce under the humps is k,ln2, which shows that only the Kra- mers doublet of Ce3+ and no Ce4+ singlet is rea- lized at low temperature, i.e. the IV phenomenon can be ruled out. 2) The Zeeman splitting E increases (linearly) with Ce concentration

X.

Extrapolated to

X =

100 at %, E is about twice the value calculated from the external field. This shows that ferromagnetic correlations are present in the paramagnetic regime.

3) The width of the Zeeman levels increases about exponentially with

X.

The extrapolation to

X =

100 at % is consistent with TK

=

(5 + ^{2) K.}

In conclusion, this TK value obtained by three different experiments on concentrated and diluted CeA1, should be considered a good measure of the dynamic single ion-conduction electron interaction energy.

3. Paramagnetic phase of CeAI,.

-

In CeA1, (with Ce on a diamond lattice), the primitive cell contains two formula units where the two Ce atoms possess sites of cubic symmetry (z3 m). The

j =

512 state of Ce3+ is CF split into a r, doublet and a T, quartet. The T, level is lower lying as is suggested by the entropy of (La, -xCex)Al, and by both resisti- vity [l31 and high field magnetization [l41 of CeAl,.

The r,-T, splitting was found by inelastic neutron scattering using cold (3.5 meV) neutrons to be 9 meV for both (La,-,Cex)A12 [IS] and CeAl, [2]. However, already in the original spectra [15, 21 an additional intensity at higher energies was seen. This was analyzed as due to magnetic scattering [16]. In a recent experi- ment performed with neutrons of 50 meV incident energy at a triple axis instrument, two inelastic lines could be clearly resolved. In figure 3, we show an energy loss spectrum for CeAl, obtained at 2 K and for Q

=

(2.5,0,0) Q, where Q,

=

2 ~18.06 A.

Besides a magnon line at I meV, which is not of interest here, two well defined lines of comparable intensity occur at 9 meV and 17.5 meV.

The existence of two inelastic lines is mysterious, it does not fit with the facts known about the system.

Fig. 3. - Energy spectrum of neutrons obtained by constant Q scan at 2 K ( k

,,,,,

⁼ 2.662

A-').

One might be tempted to think of a transition between Ce3+ and Ce4+, but since this is an ionization process no sharp transition is to be expected. Also a binding energy of 200 K for the 4f level is too small

:

suscep- tibility measurements on CeA1, up to 1 000 K can be quantitatively explained by the 512 and 712 states of the Ce3+ ions without any admixture of a Ce4+

state [17]. A new check on the crystallographic pro- perties of CeA1, did not give any indication (within lO-') of a static distortion [18]. So, one way out seems to presume a dynamic Jahn Teller effect in the excited r, quartet [19].

From a strong increase of quasielastic line width above 60 K [l51 we infer that the (Kondo type), relaxation of Ce is considerably faster in the excited CF level than in the ground state doublet. Such an increase of line width can be qualitatively understood already from second order perturbation theory with respect to the exchange coupling [20]. However, in accordance with the interpretation of previous specific heat [21] and resistivity [22] measurements, we think that the dynamic (Kondo) phenomenon is also present at high temperature.

The magnon line in figure 3 will be discussed in

the next section. Precursive ordering effects can,

however, be clearly seen in the neutron spectra of

C5-304 F. STEGLICH, C. D. BREDL, M. LOEWENHAUPT AND K. D. SCHOTTE

CeAl, far above the phase transition temperature (3.9 K). In figure 4, the intensity of the quasielastic magnetic line, IQ,, is shown as a function of Q at two different temperatures, i.e. 12 K and 80 K.

Since these data were obtained with a diffuse scatter- ing apparatus, the Q-axis does not represent a fixed direction in the crystal, but Q follows a complicated path in the reciprocal space. Therefore, only absolute Q-values are considered.

from Xc (T=12K) A T=80K T=12

K

Fig. 4. - Integrated intensity of the quasielastlc magnetic scatter- ing (corrected for 4f1 form factor) as a function of Q at two diffe- rent temperatures. Q = 0 values were calculated from the Curie part of the static susceptibility [26] (van Vleck term subtracted).

Each point in figure 4 is the average of four counters, indicated by the horizontal bars. The vertical bars represent the statistical error of the experiment.

The trivial Q-dependence due to a single ion 4f1 form factor is already eliminated in figure 4. The single ion case (no magnetic correlations) is then given by a Q-independent quasielastic intensity which agrees with x;(O,O, T).T. Apparently, this is realized at 80 K. However, at 12 K the Q-dependence of IQ, deviates considerably from uncorrelated single ion behavior, though the system is already far above its ordering temperature. In the available Q-range (0.3 to 2.5 A-'), IQ, varies by nearly a factor of 2 and shows a complicated structure pointing to precursive antiferromagnetic correlations.

Recently, spin wave like excitations were observed in the paramagnetic regime (up to T 3 T,) of MnO [23] which is well known as a Heisenberg type I1 antiferromagnet in an fcc lattice. Despite the sinu- soidal spin modulation, CeA1, shows the same structure in the ordered phase [l]. The observed precursor effects seem therefore to be inherent in this kind of antiferromagnetism which originates in the frustration of magnetic moments on face centers

when those at the cube edges tend to establish a simple type I antiferromagnet.

In addition to the structure of IQE(Q) we observe a strong discrepancy between our I* (Q

+

0) extra- polated value and that calculated from the static susceptibility. That means, in a small angle scattering experiment we would expect to see a steep increase in quasielastic intensity when reaching Q

=

0. This is a clear indication that some ferromagnetic corre- lations compete with the antiferromagnetic short range order (inferred from the IQ, variation at finite Q). A rough estimate gives a lower bound of the ferromagnetic correlation length, namely 5

>, 20

A.

In figure 5, the temperature dependence of this deviation is shown between 4 K and 80 K

:

the Curie part of the static susceptibility is compared to the integrated intensity of the quasielastic neutron scattering (averaged over 0.2

,<

QIW

^-

1.0 and extrapolated to Q

=

0). Within experimental reso- lution, both quantities agree for T >, 50 K. Only when this high temperature

(1.

12 TN) is reached, thermal fluctuations overcome magnetic correlations and CeAl, shows single ion behavior. Below 50 K, the tendency toward ferromagnetism increases, but finally the onset of long range antiferromagnetism reduces the ferromagnetic correlations. We notice that the latter are still of considerable size at 6 K where the high field Schottky hump appears in the specific heat. As is discussed above, from these humps (showing too large a Zeeman splitting), we have first inferred the existence of ferromagnetic short range ordering [3]. Croft et al. [24] arrive at a similar conclusion from an analysis of their thermal expansion measurements in high fields.

Fig. 5. - Integrated intensity of the quasielastic magnetic scatter- ing (averaged over 0.2

<

Q/A-'

<

1.0 and extrapolated to Q = 0 with single ion 4f1 form factor) as a function of temperature. Solid curve : Q = 0 values as in figure 4.

4. Antiferromagnetic phase of CeAl,.

-

As was

already mentioned in the Introduction, Barbara

et al. [l] have detected a magnetic superstructure

in their neutron diffraction pattern of CeAl, below

3.9 K. They found a type I1 antiferromagnetic struc-

ANTIFERROMAGNETIC ORDERING BETWEEN UNSTABLE 4f SHELLS IN CeAl, C5-305

ture like that of MnO, but with a magnetic propagation vector Q/Q,

=

(0.612, 0.388, 0.5) from which the polarization axis [Ill], and a modulation axis, [ I ~ o ] , are inferred. This sinusoidal modulation is incom- mensurable with the lattice. The maximum moment (0.89 f 0.05) p,, is larger than the saturation moment of the free T , level (0.71

p,),

presumably due to exchange mixing of the excited T , level into the ground state.

The first evidence for magnetic order in CeAl, was found by the observation of a magnon line and the apparent suppression of diffuse magnetic scatter- ing at 2 K [2]. Figure 6a and b show the energy dis- tribution of scattered neutrons (at 2 K) as deduced from the TOF spectra for two different Q-windows.

In contrast to the elastic nuclear line, the inelastic magnon line is Q dependent and appreciably broader than the experimental resolution. Its position varies in the available Q range (0.2 to 2.4 A-') ^between

0.8 and 1.3 meV energy transfer. At present, we have no explanation for the magnon line width (HWHM) of approximately 0.4 meV.

Fig. 6. -Energy loss spectra (incident neutron energy : 3.53 meV) of CeAl, at 2 K for two different Q windows. Solid line : Fit imply- ing one (nuclear) elastic and one (magnetic) inelastic peak. Dash- dotted line : Same fit but without folding with instrumental reso- lution function.

We notice that the spectra of figure 6 can be equally well fitted, if one allows for a quasielastic line in addition to the nuclear elastic and the magnon inelastic line. This quasielastic line would be of the same width as the inelastic one, but only of 10-20 %

intensity. A more detailed report on the magnon measurements will be given elsewhere [25].

There has been great effort to further characterize the antiferromagnetic state of CeAl,, mainly by high field magnetization [14], high field specific heat [3], and high field thermal expansion [24]. The specific heat anomaly at T,

=

3.9 K (figure 7) has typical mean field character, however with a peak height about 30 % too small and quite a broad high tempe- rature tail again indicating short range order well above T, (the entropy at TN is about 0.5 kBln2 and does not reach k,ln2 below 15 K [3]).

Fig. 7. - Specific heat of CeA1, vs. temperature. Dots : after Ref. [29]. Solid line : after Ref. [3].

10 C (JIKmole)

1

1 0-'

1 o - ~

Magnetic anisotropies must be considered in the ordered state

:

whereas at high magnetic fields the [ I l l ] direction is magnetically easiest as it is in the paramagnetic state, the [l 101 direction becomes the easiest one below 4 tesla [14, 261. At low magnetic fields the antiferromagnetically ordered spins are only slightly tilted out of their preferred [ I l l ] direc- tion. The largest tilt angle and therefore highest magnetization is achieved if the applied field has a direction perpendicular to the preferred one, that is [iio].

A step in the M(B) curves at 1.7 K and B

2.

5 tesla occurs if the field is applied along the (hard) [l001

I I

Ce A I 2

- ^-Bred1 e t a l .

t .*

-

.a*

.

.-

C

.** 2

.'

-

a-..

^-** -

~ ~ : f l + ~ ~ ~

/.

/'

/.

/'

C5-306 F. STEGLICH, C. D. BREDL, M. LOEWENHAUPT AND K. D. SCHOTTE

axis [14]. Also, specific heat anomalies in this case are much sharper than at zero field [3]. This hints at a spin flop first order transition which seems to occur if at lower temperatures a magnetic field of sufficient magnitude is applied outside the easy direction.

The size of the observed thermal expansion of polycrystalline CeAl, [27, 241 rules out the possi- bility that the magnetic phase transition is accom- panied by a structural one

:

there is only a very small homogeneous volume strain while a considerable distortive strain has been observed [24]. The latter is assumed [24] to be along the [l 1 l] axis (trigonal) which could also explain the observed decrease of the c,, elastic constant when (in the paramagnetic state) T

+ TN

1281.

We now come back to the specific heat results of figure 7. The data above 1 K are those measured by Bredl et al. [3]

;

the low temperature

data, recently obtained by Armbriister and Ste- glich [29], can be decomposed into

where y

=

130 mJ/K2 mole, fi

=

142 mJ/K4 mole, and cc

=

6.4

X

10-4 mJ K/mole. The y and /3 values are very close to those found by Bredl et al. for T > 0.3 K [3]. The PT3 term corresponds to anti- ferromagnetic magnons and allows an estimate of the interaction energy of the single Ce ion with its (four) nearest neighbors, i.e. W,,

^E

2 meV. This is in reasonable agreement with the magnon excitation energy of order l meV (see figure 6 ) .

The excess specific heat C,

= a T -

becomes relevant below 80 mK and was attributed to a hyper- fine splitting of the 27A1 nuclei transferredvia conduc- tion electron polarization from the ordered Ce moments [29]. Both, from the apparent absence of a second specific heat anomaly (at

B =

0) and from the magnitude of the transferred hyperfine field we conclude that no change in the magnetic ordering takes place down to 20 mK. According to recent measurements of the y ray anisotropy of 137Ce by Benoit et al. [30], this statement seems to hold even at T

=

5 mK.

Among the different contributions to the specific heat in (2), the yT term is the most interesting one.

It is extremely large for a magnetically ordered metal and, in addition, does not change much in high magnetic fields [3]. Therefore, it cannot be caused by a spin glass type phenomenon [31], but instead indicates a high electronic density of states a t the Fermi level. The

y

coefficient of CeAl, is more than one order of magnitude larger than that of LaAl,

[9].

We think this must be explained by a Fermi liquid effect. Since the very small volume change [27, 241 and the entropy per Ce (k,ln2) below 15 K, and also

the above mentioned resistivity measurements under pressure [4] rule out the IV phenomenon, we are left with the assumption that a Kondo effect is operative below the phase transition temperature. This was also concluded from extrapolating the specific heat results of (La, -,Ce,)Al, to

^X =

1 (section 2).

The Kondo effect characterizes an instability not in the occupation number but in the angular momen- tum of the 4f shell by the dynamic coupling to the conduction electrons. It is really surprising that this phenomenon coexists with magnetic ordering between the Ce's.

In a formal way, this situation can be treated by a mean field acting on the T7 levels of Ce which are broadened by the Kondo effect. Using the above mentioned experimental parameters for TK and

y ,

one finds TN

=

7.5 K [3]. Since mean field theory usually gives ordering temperatures too high, this result can be considered to agree well with the experi- mental value, TN

=

3.9 K (the more, because aniso- tropies and SDW have been neglected in this model).

Another interesting aspect of this phenomeno- logical treatment is concerned with the size of the

y

value. If one could switch off magnetic order, one would expect a single ion value

y =

1 740 mJ/K2 mole.

This shows that in the presence of magnetic order only about 8 % of the conduction electrons (in average) take part in the Kondo interaction. This corresponds strikingly to the 10 % reduction of the average Ce moment in the ordered phase,

ii =

(0.63

f

0.04)

p,

[l]

compared to 0.71

p,

of the free T, level. Also, a small quasielastic contribution to the neutron scattering below T, as discussed abvve would well fit into this picture.

In a similar model, Benoit et al. [32] have implied also the crystalline anisotropy of the polarized T7 doublet. Taking into account nearest neighbor ferro- magnetic and second nearest neighbor antiferro- magnetic Ce-Ce correlations they can arrive in a spin structure which is a sinusoidal one and not a helical one in agreement with experiment [l].

5. Summary and outlook.

^-

CeAl, appears to be the first metal for which coexistence of long range magnetic order and Kondo effect has been experi- mentally confirmed. A phenomenological model exists which is able to explain this surprising observation.

Several questions, however, remain to be answered

:

Why does CeAl,

^-

in contrast to all other RE)^

⁺

AI, compounds

-

order antiferromagneti- cally

?

Does ferromagnetism, which is expected to occur in CeAI, at

T, =

5 K from measurements on (Pr, -,Ce,)Al, alloys [33] and perhaps also from the above mentioned precursor effects, become pre- vented because of the Kondo effect? Or do we simply deal with the effect that indirect exchange and direct exchange couple the moments differently ?

Why are coherence effects so relatively unimportant

ANTIFERROMAGNETIC ORDERING BETWEEN UNSTABLE 4f SHELLS IN CeAl, C5-307

vis-a-vis the successful treatment of CeAl, with a of complex nature. Whether magnets of the actinides

local model

?

and of the iron group metals could be understood

Answering these questions will substantiate the in a similar way is an open question.

understanding of CeAl,. Search for other coexistence compounds will probably show that the case of

CeA1, is not singular (CeIn, appears to be a good Acknowledgments.

-

We wish to thank S. Horn candidate [34]). It could well be that CeAl, will once for his help with the neutron scattering experiments serve as an example showing how well defined magne- and B. R. Coles, P. EnteI, B. D. Rainford, U. Schotte tic moments bring about band-like magnetism, also and R. Tournier for discussions.

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