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Crystal-field effects and f-derived specific heats in heavy-fermion compounds
Journal of Magnetism and Magnetic Materials 76 & 77 (1988) 105-111 North-Holland, Amsterdam
105
INVITED PAPER CRYSTAL-FIELD EFFECTS AND f-DERIVED S P E C I F I C H E A T S IN H E A V Y - F E R M I O N C O M P O U N D S
H. RIETSCHEL, B. R E N K E R Kernforsehungszentrum Karlsruhe, D-7500 Karlsruhe, Fed. Rep. German),
R. FELTEN, F. S T E G L I C H and G. W E B E R Technische Hochschule Darmstadt, D-6100 Darmstadt, Fed. Rep. Germany
We discuss experimental results of calorimetry on heavy-fermion compounds (HFC) with special regard to the f-derived specific heat AC. Attention is given to the extraction of AC and the uncertainties implied therein. The discussion includes some Ce-based 4f systems but concentrates on U-based 5f HFC. In four of these (UBe~3, UPt 3, URuzSi 2, UzZnl7 ) distinct anomalies in AC have recently been found in the temperature range between 20 and 100 K which we think are likely to be caused by crystal-field effects thus indicating strong 5f localization.
1. Introduction The superconducting ground-state of heavyfermion (HF) compounds has been subject to numerous investigations, but much less effort has been put into clarifying their normal ground-state, although its knowledge is an indispensible prerequisite for any physical understanding of these systems. Regarding the Ce-based H F compounds like CeCu2Si 2 or CeA13, it is generally believed that the normal ground-state is that of strongly localized 4f-electrons interacting with spd conduction-electrons [1]. The H F properties are then assumed to emerge from a Kondo resonance of width k ~ T K building up in the electronic densityof-states (DOS) at E F for T_< T K (TK: Kondo temperature). By contrast, for the actinide-based H F compounds like UBe13, UPt 3 or URu2Si2, the normal ground-state is still a matter of controversy: Some researchers believe a single-particle description with fairly narrow 5f-bands at E v to be adequate [2-4], in which case the origin of the giant masses is left completely in the dark. Others favor the point of view that in these compounds, the 5f-electrons are strongly localized and highly correlated, and that the H F properties are caused in much the same way as in the Ce-based 4f systems [5]. One of the quantities which are particularly suited to discriminate between these two contrasting situations is the f-derived specific heat AC. While for itinerant band electrons, AC obeys a
linear relationship AC - 7T, for strong localization one expects sharp f-levels which are split by the crystal field (CF) thus leading to Schottky anomalies in AC. And this is what is actually observed in the Ce-based H F compounds (see also below). Recently, the observation of Schottky-type anomalies in AC has also been reported for the U-based H F compounds UBe13, URu2Si2, UPt 3 [5,6], and U2Zn17 [7] which may be taken as a strong indication of localized 5f levels split by the CF. In this contribution, we review briefly specificheat measurements on 4f and 5f H F compounds with particular regard to CF effects and to the information they may provide on the normal ground-state. Thus, measurements in the superconducting state will not be discussed. Emphasis will be laid on the extraction of the electronic contributions from the measured total specific heat and on the uncertainties involved therein. The discussion will be illustrated by experimental resuits from both our own and other laboratories. Any details regarding sample preparation and characterization and the experiments themselves may be found in the literature quoted.
2. Extraction of the f-derived specific heat Before discussing experimental results, we should like to focus attention upon the determination of AC. AC is defined by AC = Coxp - Cph as
0304-8853/88/$03.50 © Elsevier Science Publishers B.V.
H. Rietschel et al. / J:derived ,wecific heats
106
the difference between the total specific heat, C~p, and the lattice contribution Cph. The most common procedure to extract AC is to determine Cpt, separately either on a reference sample without f-electrons (e.g., LaA13 instead of CeA13) or from the phonon DOS as measured in neutron-scattering experiments and to subtract it from Q~p. Since with increasing 7", C~p is quickly dominated by ('ph, AC represents the small difference between two large numbers, and the absolute systematic error in AC is given by the sum of the absolute systematic errors in Ce~p and Cph. This may imply disastrous consequences for the relative error in AC. We consider this so important a point that we want to illustrate it by simple simulations. In these, Cph is modelled by Cph for UPt 3 as calculated from the phonon DOS [8] (any other realistic choice would do the same job) while AC is assumed to result from a sixfold degenerate f-level split by the crystal field into a doublet ground-state and a quartet 150 K higher. In a first example we now simulate a situation where the reference compound has indeed the identical Cph but where both C~p and Cp. are afflicted by systematic errors of +1% (a rather optimistic guess!), while in a second example, we simulate a situation where the experimental accuracy is perfect but where the phonon mean frequencies of the reference compound may differ from the original ones by -+ 5%. The implications of these two types of uncertainties are studied best by considering the molar entropy ~XS/R arising from AC (see fig. 1): A perfect subtraction procedure leads to 2xS/R converging towards in 6 (solid line), for systematic errors of +1% we find l n 3 . 5 < A S / R < I n 7
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(broken lines) and for +5% uncertainties in the phonon frequencies In 2.5 < A S / R < In 11 (dotted lines) at T = 200 K. So, within these fictitious experiments, any attempt to extract the degree of degeneracy, 2 J + 1, from the calorimetric data, is completely hopeless! Actually, a real situation may be more favorable, in particular for low CF splitting and high Debye temperature 0 D as in many of the Ce-based compounds. On the other hand, for the U-based HF compounds with larger CF splitting and lower 0 D, we think our simulations to be typical (see below). A rather sophisticated method to minimize the experimental uncertainties in AC has been developed by Felten et al. [6]. In order to extract A(" for UB]~, they measured Cph for both UBe13 and the reference compound ThB%3. On the same samples, they determined the phonon DOS by means of inelastic neutron scattering and calculated Cph for both compounds. Then they corrected C~xp(ThBe~3) in multiplying it by the ratio C p h ( U Be13 )/Cph(ThBe~3 ) before subtraction, thus
H. Rietschel et al. / f-derived specific heats
leading to reduced systematic errors. The result for AC is shown in fig. 2b and discussed in section 4.
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3. Cerium-based compounds In virtually all Ce-based H F compounds, the valence state of Ce is known to be Ce 3+ with an electronic 4f 1 ( J = 5/2) Hund's rule configuration. Depending on the symmetry of the crystal field, this sixfold degenerate level is split into either three doublets or into one doublet and one quartet, a magnetic doublet always being lowest level. At sufficiently low temperatures (T_< TK), the degeneracy of this doublet is then lifted by the Kondo effect thus leading to the well-known H F effects as, among many others, the appearance of a specific-heat anomaly centered around T K with molar entropy AS/R = In 2. In addition to this anomaly, one may observe a Schottky-type anomaly in AC which is due to CF excitations and which raises AS/R to the total of in 6 at high T. In some cases, the magnetic moments of the lowest doublets will order antiferromagnetically before they are fully quenched by the Kondo effect, and a X-anomaly will interfere with the Kondo anomaly, but again the total entropy will remain R In 6 at high T. When the splitting 8 between the lowest doublet and the first excited CF level becomes comparable to T K, AC can no longer be represented by a direct superposition of a Kondo (or X) and a Schottky anomaly. The separate peaks now merge, and for ~ / T K ~< 1, only one broad peak is left: The next higher level now participates in the Kondo scattering, and any separation into Kondo and Schottky anomalies has become impossible. But while this phenomenon greatly complicates data evaluation, it is well understood theoretically [9] and does not constitute any basic problem. In fig. 2a, we show AC for the "archetype" of H F compounds, CeCu2.2Si 2. In these measurements, LaCu 2.2Si 2 served as a reference to provide Cph [10]. The solid line in this figure represents the superposition of a Kondo ( T K = 13 K) and a Schottky anomaly, the latter calculated for a doublet-quartet splitting with 8 = 360 K. From neutron-scattering experiments, Horn et al. [11] deduced a somewhat different level scheme with two doublets at 140 and 360 K, respectively, in-
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stead of the quartet. This level scheme has been called into question since it is incompatible not only with the specific heat results [10] but also with results for the magnetic susceptibility [12]. Recently, Ziegler [13] has performed numerical simulations of the measurements presented in ref. [11] and shown that these data alone are insufficient to determine unambiguously the CF levels. In any case, further magnetic neutron-scattering experiments are needed to settle this question. In fig. 3 we show AC and AS/R for CeRu2.t6 Ge 2. Here, LaRu2.16Si 2 served as a reference to provide Cph [14]. There are two prominent features in AC: A X anomaly around 7 K which is magnetic in origin, and a Schottky anomaly centered around 220 K. This latter is excellently fitted by a CF level scheme with three doublets at 0, 500 and 750 K, respectively (solid line). Apparently, there is almost no interference between the low-lying anomaly and the Schottky peak. As a consequence, AS/R exhibits a pronounced plateau with height In 2 followed by a steep rise towards In 6. Thus, fig. 3 represents an outstandingly clear-cut example for separable CF effects. In fig. 4 we show AC and AS/R for C e C u 6 , one of the extreme H F compounds (y-=1.5 J / m o l K 2 ) . Again, Cph was determined on the
H. Rietschel et aL / f - d e r i v e d specific heats
108
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Ce compounds is localized so strongly that it does virtually not contribute to phonon renormalization. Measurements of the phonon DOS in CeCu 2Si2 and LaCu2Si2, respectively, corroborate this point of view [17]. But for the U-based H F compounds, the situation is likely to be different because of the lower degree of localization for the 5f electrons.
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corresponding reference compound LaCu 6 [15]. These results should be contrasted with those in fig. 3: There is only one wide anomaly with maximum around 30 K, but from the entropy which at T = 100 K is already close to R In 6, we conclude that all six 4f-states should be involved. Nevertheless, any attempt to fit AC by a superposition of a Kondo peak (T K - 7 K) and of a Schottky anomaly fails: While two excited doublets at 60 and 80 K, respectively, lead at least to the right position of the maximum, the height of the anomaly is always grossly overestimated. Obviously, we are now dealing with the above-mentioned situation of strong interference between Kondo scattering and CF excitations [9], and consistently, A S / R does not show any intermediate plateau but converges continuously towards In 6. Incidentally, the H F compound CeA13 shows very similar behavior [16]. In view of the considerations of the foregoing section, it is striking how accurately the 4f-derived entropies converge towards In 6 (see figs. 3 and 4). Obviously, the corresponding La compounds are excellent reference c o m p o u n d s with lattice specific-heats very close to that of their HF counterpart. Probably, the additional 4f electron in the
For the U-based HF compounds, the situation is much more involved. First of all, it is not clear whether the 5f electrons in these compounds are to be considered itinerant and describable in terms of conventional L D F A bandstructure-calculations [2-4], or whether they are strongly localized thus eluding any single-particle description [5]. Second, the valence state of U is not clear from the start and finally, even if we knew that we are dealing with, e.g., a 5f 3 (U -~ ) configuration, it would by no means be clear whether the three 5f electrons obey LS-coupling or whether an intermediate-coupling scheme is more appropriate [7]. Nevertheless, by now we know at least four U-based HF compounds where distinct anomalies in the 5f-derived specific heat have been found in the temperature range 20 K < T_< 100 K thus suggesting CF excitations of strongly localized 5f electrons. These compounds are UBe~ [6], URu~Si~ [5,14], UPt~ [5], and U2ZnI7 [7]. Felten et al. [6] determined AC for UBet~ in the temperature range 20 K ~< T < 120 K using the procedure described in section 2. The data show a striking similarity to those for CeCu2Si2 (see figs. 2a, b) which led the authors to interpret them in terms of a 5f 3 ( J = 9 / 2 ) configuration split by the CF into a F6-doublet as lowest level and two excited I'u quartets at 180 K and above 1000 K, respectively. Recently, Cox [18] has proposed a quadrupolar Kondo effect to explain the 5f-derived specific heat in UBe~3. While his proposal is also based on the assumption of strongly localized 5f states split by CF, it differs from the model of Ref. 6 in the CF level-scheme: A stable 5f 2 ( J - 4) Hund's-rule ground-state is now assumed with a nonmagnetic F3 doublet as lowest level. This model has the virtue of giving a better fit to the low-temperature susceptibility and provides a natural explanation for the fact that so far, no quasielastic
109
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scattering has been found in inelastic neutronscattering experiments on UBe13. At this point, a comment on such scattering experiments is in order. Goldman et al. [19] performed magnetic neutronscattering experiments on UBe13 and interpreted their results (fig. 5) as a wide ( F - 140 K) quasielastic intensity (solid line) due to spin fluctuations. These data could as well be interpreted in terms of CF excitations according to the model of ref. [6], but then the magnetic F6 doublet should give rise to additional narrow quasielastic scattering of width F - T K - 1 0 K. Estimating the strength of this scattering via the measured susceptibility X one arrives at a total cross-section corresponding to p2ff_ 1ff2, a value which can be safely excluded by the data shown in fig. 5. However, as already pointed out by Renker et al. [20], this kind of estimate which exhausts the total susceptibility X via a sum rule is erroneous: quasielastic scattering is linked only to the Curie part of X- Since at low temperature, the Van Vleck susceptibility Xvv as deduced from the CF levels proposed in ref. [6] amounts to about 75% of the total measured susceptibility x ( T = 0), only about 25% of x ( T = 0) is linked to the strength of quasielastic scattering. A more detailed calculation results in a quasielastic intensity as shown by the hatched peak in fig. 5. Obviously, such weak intensity cannot be excluded by the data of ref. [19]. Thus, the possibility that in UBel3, the lowest CF level is a magnetic doublet, is still open. We now proceed to UPt 3. By the end of 1986, AC for UPt 3 was known only for temperatures
T_< 25 K in which range AC was well described by AC = y T + 8 T 3 In T/T~ and the T 3 In T term assigned to spin fluctuations [21]. In the temperature range above 25 K, AC could not be determined since so far, nobody had succeeded to produce an isostructural reference compound ThPt 3. Recently, Renker et al. calculated Cph from the phonon DOS of UPt 3 [8] and subtracted it directly from C~xp [5]. The resulting AC is shown in fig. 6. In this subtraction procedure, the individual relative errors in Cph and C~xp where estimated to be _+1% leading to the quickly increasing error bars in AC(T). According to what has been outlined in section 2 (see fig. 1), no definitive statement regarding to entropy and thus the spin degrees-of-freedom can be deduced from this AC, but the existence of a m a x i m u m at T - 2 3 K in AC is still significant. This m a x i m u m cannot be fitted by a T 3 In T term, and AC(T) has to be reinterpreted. One possibility would be that this indicates some fine-structure in a one-particle 5f band, an argument which is supported by the fact that conventional L D F A bandstructure calculations are able to describe the Fermi-surface topology as determined in d H v A experiments [4]. But very recently, on the basis of a renormalized bandstructure calculation, Parlebas et al. [22] have shown that under certain conditions, standard L D F A theory may reproduce correctly the FS topology even for strongly localized 5f electrons. Thus, we think it more likely that also in UPt 3, we are dealing with a localized 5f configuration so that the peak reflects combined Kondo and CF
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excitations. In fact, AC is very close to the Bethe ansatz including CF splitting between two doublets as obtained in ref. [9] for the case 6 / T K = 1 with T K -= 80 K (see fig. 6). The third U-based HF compound to be discussed is URu2Si > Renker et al. [5] extracted AC from the total C~p by subtracting the specific heat for the reference compound ThRu 2Si2. The resulting data are shown in fig. 7. In addition to the ), anomaly which was first discovered by Palstra et al. [23] and ascribed to a magnetic phase transition, there is a Schottky-type anomaly centered at T - 32 K which for T < 120 K is well described by a C F excitation between two doublets separated by 3 = 75 K (solid line in fig. 7). Above 120 K, the data are increasingly affected by statistical scatter. Walter et al. [24] performed inelastic neutronscattering experiments on URu2Si 2 and found a clear inelastic magnetic transition at low temperatures with AE - 73 K. From a detailed analysis of their scattering results (but without knowledge of the more recent calorimetric data!), these authors excluded the possibility of CF excitations. However, we think that the almost perfect agreement between 3 = 75 K and AE - 73 K and in particular the typical shape of the specific-heat anomaly strongly corroborate the assumption of CF excitations in WRu2Si 2. Finally we want to mention the recent calorimetric results of Fischer et al. [7] on U2Zn17. With the help of C~p from the reference compound Th2ZnlT, they extracted AC for temperatures up to 70 K and found a specific-heat anomaly with maximum around 50 K. A fit of a CF level-scheme gave only moderate agreement while a DOS model with two narrow ( - 5 0 K) peaks with a 108 Ksplitting described the data satisfactorily. From
this and their overshooting entropy which at T 70 K already exceeds R In 11, the authors conclude that some intermediate-coupling scheme or even band scheme should be a more appropriate description of the 5f electrons in U2Zn~7 than a CF scheme. However, we think that these latter conclusions cannot be drawn from the data shown in ref. [7] since the given estimate of errors seems much too optimistic to us. In particular the overshooting entropy should not be considered significant since, as has been outlined in section 2, already a modest difference between the phonon spectra for U2Znlv and Th2Znlv, respectively, may explain large discrepancies in 2 J q I (see fig. 1, broken lines!). Experimental determination of the phonon DOS in U:Zn w and Th2Zn w, respectively, would be highly desirable in order to allow for a more conclusive evaluation of the calorimetric data. 5. Conclusions In this article, we discussed selected measurements of the f-derived specific heat in heavy-fermion compounds with the particular aim of deriving information on possible C F effects and thus on the degree of localization of the f electrons in these systems. For the Ce-based HF compounds, we obtained the rather unified picture of strongly localized 4f electrons with pronounced specificheat anomalies which can be traced back unambiguously to CF effects. So, for the theoretical description of the physics of these compounds, the Anderson-lattice Hamiltonian should be the appropriate starting point, and the HF properties are likely to emerge from the Kondo effect. For the U-based HF compounds, the situation is less clear. The lower degree of localization for the 5f electrons opens a priori the alternative that these are occupying narrow bands close to Ej., a hypothesis which is often fiercely defended by the bandstructure groups, although in that case the origin of the HF properties remains a puzzle. The fact that already in four of these compounds (not many more are known!), distinct specific-heat anomalies have been detected, makes it appear likely that we are dealing with strongly localized 5f electrons and possibly with CF effects, a conjecture which finds strong support also from experimental results for other non-calorimetric data
H. Rietsehel et al. / f-derived specific heats
as, e.g., lattice d y n a m i c s [5] or m a g n e t i c b e h a v i o r [25]. H o w else s h o u l d o n e e x p l a i n the l o w - l y i n g (-100 K ) e l e c t r o n i c e x c i t a t i o n s in these c o m p o u n d s if n o t b y s t r o n g l y l o c a l i z e d ( a n d thus s t r o n g l y c o r r e l a t e d ) 5f e l e c t r o n s ? A d e f i n i t i v e a n s w e r , h o w e v e r , to these q u e s t i o n s n e e d s a d d i t i o n a l e x p e r i m e n t a l results w h e r e high-accuracy calorimetric measurements combined with inelastic neutron-scattering experim e n t s will p l a y a c e n t r a l role. W e h o p e t h a t at least s o m e o f t h o s e g r o u p s w h o h a v e left this field to share the a d v e n t u r e of high-T~ s u p e r c o n d u c t i v ity, will c o m e b a c k to c o n t i n u e this b e a u t i f u l physics.
References [1] C.D. Bredl, S. Horn, F. Steglich, B. Ltithi, and R.M. Martin, Phys. Rev. Lett. 52 (1984) 1982. [2] A.M. Boring, R.C. Albers, T.M. Mueller and D.D. Koelling, Physica B130 (1985) 171. [3] T. Oguchi, A.J. Freeman and G.W. Crabtree, J. Magn. Magn. Mat. 63&64 (1987) 645. [4] C.S. Wang, M.R. Norman, R.C. Albers, A.M. Boring, W.E. Picket, H. Krakauer and N.E. Christensen, Phys. Rev. B35 (1987) 7260. [5] B. Renker, F. Gompf, E. Gering, P. Frings, H. Rietschel, R. Felten, F. Steglich and G. Weber, Physica B 148 (1987) 41. [6] R. Felten, F. Steglich, G. Weber, H. Rietschel, F. Gompf, B. Renker and J. Beuers, Europhys. Lett. 2 (1986) 323. [7] H.E. Fischer, E.T. Swartz, R.O. Pohk B.A. Jones, J.W. Wilkins, and Z, Fisk, Phys. Rev. B34 (1987) 5330. [8] B. Renker, F. Gompf, J.B. Suck, H. Rietschel and P.H. Frings, Physica B 136 (1986) 376.
111
[9] H.-U. Desgranges and J.W. Rasul, Phys. Rev. B32 (1985) 6100. [10] C.D. Bredl, W. Lieke, R. Schefzyk, M. Lang, U. Rauchschwalbe, F. Steglich, S. Riegel, R. Felten, G. Weber, J. Klaasse, J. Aarts and F.R. de Boer, J. Magn. Magn. Mat. 74&48 (1985) 30. [11] S. Horn, E. Holland-Moritz, M. Loewenhaupt, F. Steglich, H. Scheuer, A. Benoit and J. Flouquet, Phys. Rev. B23 (1981) 3171. [12] S. Maekawa, S. Kosbiba, S. Takahashi and M. Tachiki, J. Magn. Magn. Mat. 54-57 (1986) 355. [13] C. Ziegler, Diploma thesis (KfK 1987) unpublished. [14] R. Felten, G. Weber and H. Rietschek J. Magn. Magn. Mat. 63&64 (1987) 383. [15] R. Felten, Thesis (TH Darmstadt 1987) unpublished. [16] F.R. de Boer, J. Klaasse, J. Aarts, C.D. Bredl, W. Lieke, U. Rauchschwalbe, F. Steglich, R. Felten, U. Umhofer and G. Weber, J. Magn. Magn. Mat. 47&48 (1985) 60. [17] F. Gompf, E. Gering, B. Renker, H. Rietschel, U. Rauchschwalbe and F. Steglich, J. Magn. Magn. Mat. 63&64 (1987) 344. [18] D.L. Cox, Phys. Rev. Lett. 59 (1987) 1240. [19] A.I. Goldman, S.M. Shapiro, G. Shirane, J.L. Smith and Z. Fisk, Phys. Rev. 33 (1986) 1627. [20] B. Renker, E. Gering, F. Gompf, H. Schmidt and H. Rietschel, J. Magn. Magn. Mat. 63&64 (1987) 31. [21] G.R. Stewart, Z. Fisk, J.O. Willis and J.L. Smith, Phys. Rev. Lett. 52 (1984) 679. [22] J.C. Parlebas, N.E. Christensen, E. Runge and G. Zwicknagel, preprint (1988), See also contribution to this Conference. [23] T.T. Palstra, A.A. Menovsky, J. van den Berg, A.J. Dirkmaat, P.H. Kes, G.J. Nieuwenhuys and J.A. Mydosh, Phys. Rev. Lett. 55 (1985) 2727. [24] U. Walter, C.-K. Loong, M. Loewenhaupt and W. Schlabitz, Phys. Rev. B 33 (1986) 7875. [25] G.J. Nieuwenhuys, Phys. Rev. B 34 (1987) 5260.
105
INVITED PAPER CRYSTAL-FIELD EFFECTS AND f-DERIVED S P E C I F I C H E A T S IN H E A V Y - F E R M I O N C O M P O U N D S
H. RIETSCHEL, B. R E N K E R Kernforsehungszentrum Karlsruhe, D-7500 Karlsruhe, Fed. Rep. German),
R. FELTEN, F. S T E G L I C H and G. W E B E R Technische Hochschule Darmstadt, D-6100 Darmstadt, Fed. Rep. Germany
We discuss experimental results of calorimetry on heavy-fermion compounds (HFC) with special regard to the f-derived specific heat AC. Attention is given to the extraction of AC and the uncertainties implied therein. The discussion includes some Ce-based 4f systems but concentrates on U-based 5f HFC. In four of these (UBe~3, UPt 3, URuzSi 2, UzZnl7 ) distinct anomalies in AC have recently been found in the temperature range between 20 and 100 K which we think are likely to be caused by crystal-field effects thus indicating strong 5f localization.
1. Introduction The superconducting ground-state of heavyfermion (HF) compounds has been subject to numerous investigations, but much less effort has been put into clarifying their normal ground-state, although its knowledge is an indispensible prerequisite for any physical understanding of these systems. Regarding the Ce-based H F compounds like CeCu2Si 2 or CeA13, it is generally believed that the normal ground-state is that of strongly localized 4f-electrons interacting with spd conduction-electrons [1]. The H F properties are then assumed to emerge from a Kondo resonance of width k ~ T K building up in the electronic densityof-states (DOS) at E F for T_< T K (TK: Kondo temperature). By contrast, for the actinide-based H F compounds like UBe13, UPt 3 or URu2Si2, the normal ground-state is still a matter of controversy: Some researchers believe a single-particle description with fairly narrow 5f-bands at E v to be adequate [2-4], in which case the origin of the giant masses is left completely in the dark. Others favor the point of view that in these compounds, the 5f-electrons are strongly localized and highly correlated, and that the H F properties are caused in much the same way as in the Ce-based 4f systems [5]. One of the quantities which are particularly suited to discriminate between these two contrasting situations is the f-derived specific heat AC. While for itinerant band electrons, AC obeys a
linear relationship AC - 7T, for strong localization one expects sharp f-levels which are split by the crystal field (CF) thus leading to Schottky anomalies in AC. And this is what is actually observed in the Ce-based H F compounds (see also below). Recently, the observation of Schottky-type anomalies in AC has also been reported for the U-based H F compounds UBe13, URu2Si2, UPt 3 [5,6], and U2Zn17 [7] which may be taken as a strong indication of localized 5f levels split by the CF. In this contribution, we review briefly specificheat measurements on 4f and 5f H F compounds with particular regard to CF effects and to the information they may provide on the normal ground-state. Thus, measurements in the superconducting state will not be discussed. Emphasis will be laid on the extraction of the electronic contributions from the measured total specific heat and on the uncertainties involved therein. The discussion will be illustrated by experimental resuits from both our own and other laboratories. Any details regarding sample preparation and characterization and the experiments themselves may be found in the literature quoted.
2. Extraction of the f-derived specific heat Before discussing experimental results, we should like to focus attention upon the determination of AC. AC is defined by AC = Coxp - Cph as
0304-8853/88/$03.50 © Elsevier Science Publishers B.V.
H. Rietschel et al. / J:derived ,wecific heats
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the difference between the total specific heat, C~p, and the lattice contribution Cph. The most common procedure to extract AC is to determine Cpt, separately either on a reference sample without f-electrons (e.g., LaA13 instead of CeA13) or from the phonon DOS as measured in neutron-scattering experiments and to subtract it from Q~p. Since with increasing 7", C~p is quickly dominated by ('ph, AC represents the small difference between two large numbers, and the absolute systematic error in AC is given by the sum of the absolute systematic errors in Ce~p and Cph. This may imply disastrous consequences for the relative error in AC. We consider this so important a point that we want to illustrate it by simple simulations. In these, Cph is modelled by Cph for UPt 3 as calculated from the phonon DOS [8] (any other realistic choice would do the same job) while AC is assumed to result from a sixfold degenerate f-level split by the crystal field into a doublet ground-state and a quartet 150 K higher. In a first example we now simulate a situation where the reference compound has indeed the identical Cph but where both C~p and Cp. are afflicted by systematic errors of +1% (a rather optimistic guess!), while in a second example, we simulate a situation where the experimental accuracy is perfect but where the phonon mean frequencies of the reference compound may differ from the original ones by -+ 5%. The implications of these two types of uncertainties are studied best by considering the molar entropy ~XS/R arising from AC (see fig. 1): A perfect subtraction procedure leads to 2xS/R converging towards in 6 (solid line), for systematic errors of +1% we find l n 3 . 5 < A S / R < I n 7
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(broken lines) and for +5% uncertainties in the phonon frequencies In 2.5 < A S / R < In 11 (dotted lines) at T = 200 K. So, within these fictitious experiments, any attempt to extract the degree of degeneracy, 2 J + 1, from the calorimetric data, is completely hopeless! Actually, a real situation may be more favorable, in particular for low CF splitting and high Debye temperature 0 D as in many of the Ce-based compounds. On the other hand, for the U-based HF compounds with larger CF splitting and lower 0 D, we think our simulations to be typical (see below). A rather sophisticated method to minimize the experimental uncertainties in AC has been developed by Felten et al. [6]. In order to extract A(" for UB]~, they measured Cph for both UBe13 and the reference compound ThB%3. On the same samples, they determined the phonon DOS by means of inelastic neutron scattering and calculated Cph for both compounds. Then they corrected C~xp(ThBe~3) in multiplying it by the ratio C p h ( U Be13 )/Cph(ThBe~3 ) before subtraction, thus
H. Rietschel et al. / f-derived specific heats
leading to reduced systematic errors. The result for AC is shown in fig. 2b and discussed in section 4.
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3. Cerium-based compounds In virtually all Ce-based H F compounds, the valence state of Ce is known to be Ce 3+ with an electronic 4f 1 ( J = 5/2) Hund's rule configuration. Depending on the symmetry of the crystal field, this sixfold degenerate level is split into either three doublets or into one doublet and one quartet, a magnetic doublet always being lowest level. At sufficiently low temperatures (T_< TK), the degeneracy of this doublet is then lifted by the Kondo effect thus leading to the well-known H F effects as, among many others, the appearance of a specific-heat anomaly centered around T K with molar entropy AS/R = In 2. In addition to this anomaly, one may observe a Schottky-type anomaly in AC which is due to CF excitations and which raises AS/R to the total of in 6 at high T. In some cases, the magnetic moments of the lowest doublets will order antiferromagnetically before they are fully quenched by the Kondo effect, and a X-anomaly will interfere with the Kondo anomaly, but again the total entropy will remain R In 6 at high T. When the splitting 8 between the lowest doublet and the first excited CF level becomes comparable to T K, AC can no longer be represented by a direct superposition of a Kondo (or X) and a Schottky anomaly. The separate peaks now merge, and for ~ / T K ~< 1, only one broad peak is left: The next higher level now participates in the Kondo scattering, and any separation into Kondo and Schottky anomalies has become impossible. But while this phenomenon greatly complicates data evaluation, it is well understood theoretically [9] and does not constitute any basic problem. In fig. 2a, we show AC for the "archetype" of H F compounds, CeCu2.2Si 2. In these measurements, LaCu 2.2Si 2 served as a reference to provide Cph [10]. The solid line in this figure represents the superposition of a Kondo ( T K = 13 K) and a Schottky anomaly, the latter calculated for a doublet-quartet splitting with 8 = 360 K. From neutron-scattering experiments, Horn et al. [11] deduced a somewhat different level scheme with two doublets at 140 and 360 K, respectively, in-
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stead of the quartet. This level scheme has been called into question since it is incompatible not only with the specific heat results [10] but also with results for the magnetic susceptibility [12]. Recently, Ziegler [13] has performed numerical simulations of the measurements presented in ref. [11] and shown that these data alone are insufficient to determine unambiguously the CF levels. In any case, further magnetic neutron-scattering experiments are needed to settle this question. In fig. 3 we show AC and AS/R for CeRu2.t6 Ge 2. Here, LaRu2.16Si 2 served as a reference to provide Cph [14]. There are two prominent features in AC: A X anomaly around 7 K which is magnetic in origin, and a Schottky anomaly centered around 220 K. This latter is excellently fitted by a CF level scheme with three doublets at 0, 500 and 750 K, respectively (solid line). Apparently, there is almost no interference between the low-lying anomaly and the Schottky peak. As a consequence, AS/R exhibits a pronounced plateau with height In 2 followed by a steep rise towards In 6. Thus, fig. 3 represents an outstandingly clear-cut example for separable CF effects. In fig. 4 we show AC and AS/R for C e C u 6 , one of the extreme H F compounds (y-=1.5 J / m o l K 2 ) . Again, Cph was determined on the
H. Rietschel et aL / f - d e r i v e d specific heats
108
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Ce compounds is localized so strongly that it does virtually not contribute to phonon renormalization. Measurements of the phonon DOS in CeCu 2Si2 and LaCu2Si2, respectively, corroborate this point of view [17]. But for the U-based H F compounds, the situation is likely to be different because of the lower degree of localization for the 5f electrons.
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corresponding reference compound LaCu 6 [15]. These results should be contrasted with those in fig. 3: There is only one wide anomaly with maximum around 30 K, but from the entropy which at T = 100 K is already close to R In 6, we conclude that all six 4f-states should be involved. Nevertheless, any attempt to fit AC by a superposition of a Kondo peak (T K - 7 K) and of a Schottky anomaly fails: While two excited doublets at 60 and 80 K, respectively, lead at least to the right position of the maximum, the height of the anomaly is always grossly overestimated. Obviously, we are now dealing with the above-mentioned situation of strong interference between Kondo scattering and CF excitations [9], and consistently, A S / R does not show any intermediate plateau but converges continuously towards In 6. Incidentally, the H F compound CeA13 shows very similar behavior [16]. In view of the considerations of the foregoing section, it is striking how accurately the 4f-derived entropies converge towards In 6 (see figs. 3 and 4). Obviously, the corresponding La compounds are excellent reference c o m p o u n d s with lattice specific-heats very close to that of their HF counterpart. Probably, the additional 4f electron in the
For the U-based HF compounds, the situation is much more involved. First of all, it is not clear whether the 5f electrons in these compounds are to be considered itinerant and describable in terms of conventional L D F A bandstructure-calculations [2-4], or whether they are strongly localized thus eluding any single-particle description [5]. Second, the valence state of U is not clear from the start and finally, even if we knew that we are dealing with, e.g., a 5f 3 (U -~ ) configuration, it would by no means be clear whether the three 5f electrons obey LS-coupling or whether an intermediate-coupling scheme is more appropriate [7]. Nevertheless, by now we know at least four U-based HF compounds where distinct anomalies in the 5f-derived specific heat have been found in the temperature range 20 K < T_< 100 K thus suggesting CF excitations of strongly localized 5f electrons. These compounds are UBe~ [6], URu~Si~ [5,14], UPt~ [5], and U2ZnI7 [7]. Felten et al. [6] determined AC for UBet~ in the temperature range 20 K ~< T < 120 K using the procedure described in section 2. The data show a striking similarity to those for CeCu2Si2 (see figs. 2a, b) which led the authors to interpret them in terms of a 5f 3 ( J = 9 / 2 ) configuration split by the CF into a F6-doublet as lowest level and two excited I'u quartets at 180 K and above 1000 K, respectively. Recently, Cox [18] has proposed a quadrupolar Kondo effect to explain the 5f-derived specific heat in UBe~3. While his proposal is also based on the assumption of strongly localized 5f states split by CF, it differs from the model of Ref. 6 in the CF level-scheme: A stable 5f 2 ( J - 4) Hund's-rule ground-state is now assumed with a nonmagnetic F3 doublet as lowest level. This model has the virtue of giving a better fit to the low-temperature susceptibility and provides a natural explanation for the fact that so far, no quasielastic
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scattering has been found in inelastic neutronscattering experiments on UBe13. At this point, a comment on such scattering experiments is in order. Goldman et al. [19] performed magnetic neutronscattering experiments on UBe13 and interpreted their results (fig. 5) as a wide ( F - 140 K) quasielastic intensity (solid line) due to spin fluctuations. These data could as well be interpreted in terms of CF excitations according to the model of ref. [6], but then the magnetic F6 doublet should give rise to additional narrow quasielastic scattering of width F - T K - 1 0 K. Estimating the strength of this scattering via the measured susceptibility X one arrives at a total cross-section corresponding to p2ff_ 1ff2, a value which can be safely excluded by the data shown in fig. 5. However, as already pointed out by Renker et al. [20], this kind of estimate which exhausts the total susceptibility X via a sum rule is erroneous: quasielastic scattering is linked only to the Curie part of X- Since at low temperature, the Van Vleck susceptibility Xvv as deduced from the CF levels proposed in ref. [6] amounts to about 75% of the total measured susceptibility x ( T = 0), only about 25% of x ( T = 0) is linked to the strength of quasielastic scattering. A more detailed calculation results in a quasielastic intensity as shown by the hatched peak in fig. 5. Obviously, such weak intensity cannot be excluded by the data of ref. [19]. Thus, the possibility that in UBel3, the lowest CF level is a magnetic doublet, is still open. We now proceed to UPt 3. By the end of 1986, AC for UPt 3 was known only for temperatures
T_< 25 K in which range AC was well described by AC = y T + 8 T 3 In T/T~ and the T 3 In T term assigned to spin fluctuations [21]. In the temperature range above 25 K, AC could not be determined since so far, nobody had succeeded to produce an isostructural reference compound ThPt 3. Recently, Renker et al. calculated Cph from the phonon DOS of UPt 3 [8] and subtracted it directly from C~xp [5]. The resulting AC is shown in fig. 6. In this subtraction procedure, the individual relative errors in Cph and C~xp where estimated to be _+1% leading to the quickly increasing error bars in AC(T). According to what has been outlined in section 2 (see fig. 1), no definitive statement regarding to entropy and thus the spin degrees-of-freedom can be deduced from this AC, but the existence of a m a x i m u m at T - 2 3 K in AC is still significant. This m a x i m u m cannot be fitted by a T 3 In T term, and AC(T) has to be reinterpreted. One possibility would be that this indicates some fine-structure in a one-particle 5f band, an argument which is supported by the fact that conventional L D F A bandstructure calculations are able to describe the Fermi-surface topology as determined in d H v A experiments [4]. But very recently, on the basis of a renormalized bandstructure calculation, Parlebas et al. [22] have shown that under certain conditions, standard L D F A theory may reproduce correctly the FS topology even for strongly localized 5f electrons. Thus, we think it more likely that also in UPt 3, we are dealing with a localized 5f configuration so that the peak reflects combined Kondo and CF
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excitations. In fact, AC is very close to the Bethe ansatz including CF splitting between two doublets as obtained in ref. [9] for the case 6 / T K = 1 with T K -= 80 K (see fig. 6). The third U-based HF compound to be discussed is URu2Si > Renker et al. [5] extracted AC from the total C~p by subtracting the specific heat for the reference compound ThRu 2Si2. The resulting data are shown in fig. 7. In addition to the ), anomaly which was first discovered by Palstra et al. [23] and ascribed to a magnetic phase transition, there is a Schottky-type anomaly centered at T - 32 K which for T < 120 K is well described by a C F excitation between two doublets separated by 3 = 75 K (solid line in fig. 7). Above 120 K, the data are increasingly affected by statistical scatter. Walter et al. [24] performed inelastic neutronscattering experiments on URu2Si 2 and found a clear inelastic magnetic transition at low temperatures with AE - 73 K. From a detailed analysis of their scattering results (but without knowledge of the more recent calorimetric data!), these authors excluded the possibility of CF excitations. However, we think that the almost perfect agreement between 3 = 75 K and AE - 73 K and in particular the typical shape of the specific-heat anomaly strongly corroborate the assumption of CF excitations in WRu2Si 2. Finally we want to mention the recent calorimetric results of Fischer et al. [7] on U2Zn17. With the help of C~p from the reference compound Th2ZnlT, they extracted AC for temperatures up to 70 K and found a specific-heat anomaly with maximum around 50 K. A fit of a CF level-scheme gave only moderate agreement while a DOS model with two narrow ( - 5 0 K) peaks with a 108 Ksplitting described the data satisfactorily. From
this and their overshooting entropy which at T 70 K already exceeds R In 11, the authors conclude that some intermediate-coupling scheme or even band scheme should be a more appropriate description of the 5f electrons in U2Zn~7 than a CF scheme. However, we think that these latter conclusions cannot be drawn from the data shown in ref. [7] since the given estimate of errors seems much too optimistic to us. In particular the overshooting entropy should not be considered significant since, as has been outlined in section 2, already a modest difference between the phonon spectra for U2Znlv and Th2Znlv, respectively, may explain large discrepancies in 2 J q I (see fig. 1, broken lines!). Experimental determination of the phonon DOS in U:Zn w and Th2Zn w, respectively, would be highly desirable in order to allow for a more conclusive evaluation of the calorimetric data. 5. Conclusions In this article, we discussed selected measurements of the f-derived specific heat in heavy-fermion compounds with the particular aim of deriving information on possible C F effects and thus on the degree of localization of the f electrons in these systems. For the Ce-based HF compounds, we obtained the rather unified picture of strongly localized 4f electrons with pronounced specificheat anomalies which can be traced back unambiguously to CF effects. So, for the theoretical description of the physics of these compounds, the Anderson-lattice Hamiltonian should be the appropriate starting point, and the HF properties are likely to emerge from the Kondo effect. For the U-based HF compounds, the situation is less clear. The lower degree of localization for the 5f electrons opens a priori the alternative that these are occupying narrow bands close to Ej., a hypothesis which is often fiercely defended by the bandstructure groups, although in that case the origin of the HF properties remains a puzzle. The fact that already in four of these compounds (not many more are known!), distinct specific-heat anomalies have been detected, makes it appear likely that we are dealing with strongly localized 5f electrons and possibly with CF effects, a conjecture which finds strong support also from experimental results for other non-calorimetric data
H. Rietsehel et al. / f-derived specific heats
as, e.g., lattice d y n a m i c s [5] or m a g n e t i c b e h a v i o r [25]. H o w else s h o u l d o n e e x p l a i n the l o w - l y i n g (-100 K ) e l e c t r o n i c e x c i t a t i o n s in these c o m p o u n d s if n o t b y s t r o n g l y l o c a l i z e d ( a n d thus s t r o n g l y c o r r e l a t e d ) 5f e l e c t r o n s ? A d e f i n i t i v e a n s w e r , h o w e v e r , to these q u e s t i o n s n e e d s a d d i t i o n a l e x p e r i m e n t a l results w h e r e high-accuracy calorimetric measurements combined with inelastic neutron-scattering experim e n t s will p l a y a c e n t r a l role. W e h o p e t h a t at least s o m e o f t h o s e g r o u p s w h o h a v e left this field to share the a d v e n t u r e of high-T~ s u p e r c o n d u c t i v ity, will c o m e b a c k to c o n t i n u e this b e a u t i f u l physics.
References [1] C.D. Bredl, S. Horn, F. Steglich, B. Ltithi, and R.M. Martin, Phys. Rev. Lett. 52 (1984) 1982. [2] A.M. Boring, R.C. Albers, T.M. Mueller and D.D. Koelling, Physica B130 (1985) 171. [3] T. Oguchi, A.J. Freeman and G.W. Crabtree, J. Magn. Magn. Mat. 63&64 (1987) 645. [4] C.S. Wang, M.R. Norman, R.C. Albers, A.M. Boring, W.E. Picket, H. Krakauer and N.E. Christensen, Phys. Rev. B35 (1987) 7260. [5] B. Renker, F. Gompf, E. Gering, P. Frings, H. Rietschel, R. Felten, F. Steglich and G. Weber, Physica B 148 (1987) 41. [6] R. Felten, F. Steglich, G. Weber, H. Rietschel, F. Gompf, B. Renker and J. Beuers, Europhys. Lett. 2 (1986) 323. [7] H.E. Fischer, E.T. Swartz, R.O. Pohk B.A. Jones, J.W. Wilkins, and Z, Fisk, Phys. Rev. B34 (1987) 5330. [8] B. Renker, F. Gompf, J.B. Suck, H. Rietschel and P.H. Frings, Physica B 136 (1986) 376.
111
[9] H.-U. Desgranges and J.W. Rasul, Phys. Rev. B32 (1985) 6100. [10] C.D. Bredl, W. Lieke, R. Schefzyk, M. Lang, U. Rauchschwalbe, F. Steglich, S. Riegel, R. Felten, G. Weber, J. Klaasse, J. Aarts and F.R. de Boer, J. Magn. Magn. Mat. 74&48 (1985) 30. [11] S. Horn, E. Holland-Moritz, M. Loewenhaupt, F. Steglich, H. Scheuer, A. Benoit and J. Flouquet, Phys. Rev. B23 (1981) 3171. [12] S. Maekawa, S. Kosbiba, S. Takahashi and M. Tachiki, J. Magn. Magn. Mat. 54-57 (1986) 355. [13] C. Ziegler, Diploma thesis (KfK 1987) unpublished. [14] R. Felten, G. Weber and H. Rietschek J. Magn. Magn. Mat. 63&64 (1987) 383. [15] R. Felten, Thesis (TH Darmstadt 1987) unpublished. [16] F.R. de Boer, J. Klaasse, J. Aarts, C.D. Bredl, W. Lieke, U. Rauchschwalbe, F. Steglich, R. Felten, U. Umhofer and G. Weber, J. Magn. Magn. Mat. 47&48 (1985) 60. [17] F. Gompf, E. Gering, B. Renker, H. Rietschel, U. Rauchschwalbe and F. Steglich, J. Magn. Magn. Mat. 63&64 (1987) 344. [18] D.L. Cox, Phys. Rev. Lett. 59 (1987) 1240. [19] A.I. Goldman, S.M. Shapiro, G. Shirane, J.L. Smith and Z. Fisk, Phys. Rev. 33 (1986) 1627. [20] B. Renker, E. Gering, F. Gompf, H. Schmidt and H. Rietschel, J. Magn. Magn. Mat. 63&64 (1987) 31. [21] G.R. Stewart, Z. Fisk, J.O. Willis and J.L. Smith, Phys. Rev. Lett. 52 (1984) 679. [22] J.C. Parlebas, N.E. Christensen, E. Runge and G. Zwicknagel, preprint (1988), See also contribution to this Conference. [23] T.T. Palstra, A.A. Menovsky, J. van den Berg, A.J. Dirkmaat, P.H. Kes, G.J. Nieuwenhuys and J.A. Mydosh, Phys. Rev. Lett. 55 (1985) 2727. [24] U. Walter, C.-K. Loong, M. Loewenhaupt and W. Schlabitz, Phys. Rev. B 33 (1986) 7875. [25] G.J. Nieuwenhuys, Phys. Rev. B 34 (1987) 5260.