Convergence of the enhancement of the effective mass under pressure and magnetic field in heavy-fermion compounds: CeRu2Si2, CeRh2Si2, and CeIn3
J. Flouquet, D. Aoki, W. Knafo, G. Knebel, T.D. Matsuda, S. Raymond, C. Proust, C. Paulsen, P. Haen
aa r X i v : . [ c ond - m a t . s t r- e l ] M a y Noname manuscript No. (will be inserted by the editor)
J. Flouquet · D. Aoki · W. Knafo · G.Knebel · T.D. Matsuda · S. Raymond · C. Proust · C. Paulsen · P. Haen
Convergence of the enhancement ofthe effective mass under pressureand magnetic field in heavy-fermioncompounds: CeRu Si , CeRh Si , andCeIn Received: date / Accepted: date
Abstract
Emphasis is given on the observation of a convergence to a criticalvalue of the effective mass of a heavy fermion compound by tuning it througha quantum instability either by applying pressure or magnetic field from anantiferromagnetic (AF) to a paramagnetic (PM) ground state. Macroscopicand microscopic results are discussed and the main message is to rush to thediscovery of an ideal material whose Fermi surface could be fully observedon both sides of each quantum phase transition.
Keywords quantum criticality · heavy fermion · pseudo-metamagnetism · CeRu Si PACS
PACS 75.30.Mb · · T F in heavy fermion compounds (HFC), it is possible totune them with moderate values of pressure ( p ) or magnetic field ( H ) froma long range antiferromagnetic (AF) ground state to a paramagnetic (PM)one at a critical pressure p = p c or a critical field H = H c [1]. Under pressure J. Flouquet · D. Aoki · G. Knebel · T.D. Matsuda · S. RaymondSPSMS, UMR-E 9001, CEA-INAC/ UJF-Grenoble 1, 17 rue des Martyrs, 38054Grenoble Cedex 9, FranceTel.: +33-43878-5423Fax: +33-43878-5096E-mail: jacques.fl[email protected]. Knafo · C. ProustLaboratoire National des Champs Magn´etiques Intenses, UPR 3228, CNRS-UJF-UPS-INSA, 143 Avenue de Rangueil, 31400 Toulouse, FranceC. Paulsen · P. HaenInstitut N´eel, CNRS / UJF Grenoble, BP166, 38042 Grenoble Cedex 9, France the main phenomena can be considered to be governed by the collapse of theAF order parameter. At low temperature and under magnetic field, often theachievement of a significant high magnetic polarization near H c ends up in apolarized paramagnetic (PPM) phase with a marked crossover on warmingfrom the low field paramagnetic phase. Often, for fields close to the criticalfield H c , the magnetic polarization of the 4 f centers reaches typically 20%of the full moment. Thus, ferromagnetic interactions must certainly playa major role. Assuming that the 4 f electrons are itinerant, the differencebetween the majority and minority spin bands should have consequences onthe Fermi surface.On the paramagnetic side of the phase diagram (see Fig. 1) the vicinityof the critical pressure p c is characterized by a large electronic Gr¨uneisenparameter Ω ( T →
0) [2,3]. The Gr¨uneisen parameter Ω ( T ) is defined by theratio of the thermal expansion α and the specific heat C multiplied with theratio of the molar volume V and the compressibility κ ( Ω ( T ) = αC × Vκ ).It has been observed that a constant value of Ω ( T ) is only approached atvery low temperatures [2,3]. Basically, the continuous increase of Ω ( T ) oncooling is a direct macroscopic evidence of a large non Fermi liquid domainin temperature. In this regime, the free energy F is not controlled by asingle energy scale T ⋆ and F cannot be reduced to a simple expression F = T Φ ( T /T ⋆ ) which would imply Ω ( T ) = Ω (0) = − ∂ log T ⋆ /∂ log V .In many HFC such as CeNi Ge [4], Ω ( T →
0) seems to diverge at p c .Figure 2 shows the temperature dependence of Ω of CeRu Si at p = 0 [1,5]. The specific interest of CeRu Si is that at p = 0 it is located slightlyabove the ”effective” negative critical pressure ( − p c = 0 . − . − x La x Ru Si series. At ambient pressure thecritical concentration at which the N´eel temperature T N vanishes is closeto x c = 0 .
075 [7,10]. On the AF side, at a concentration x = 0 . T N =6 K), sweeping the magnetic field at zero pressure leads to two successivemetamagnetic transitions at H a and H c , corresponding to phase transitionsbetween two AF structures (at H a ) and between AF and PM phases (at H c ) [7]. Furthermore, for this La concentration x = 0 .
2, antiferromagnetismcollapses at a pressure p c = 0 . H ⋆c = 3 . H ⋆c is felt by the occurenceof a sharp pseudo-metamagnetic transition at H M (see Fig. 3). The aim ofthe present article is to focus on the interplay between the pressure and fieldinstabilities with a special emphasis on the mass enhancement at the criticalpressure, m ⋆ ( p c ), or at the critical field, m ⋆ ( H c ) or m ⋆ ( H M ). Our attentionthat the mass enhancement may be the same at the critical points p c , H c or H M emerged from recent measurements made on CeRh Si which is a HFCsituated deep inside the AF phase at p = 0 ( T N = 36 K, H c = 26 T) [12],but where AF order is rapidly suppressed under pressure. For CeRh Si a PM ground state is realized already at p c ≈ m ⋆ ( H c ) and m ⋆ ( p c ) can be pointed out from former studiesof heavy fermion systems, as that of CeIn either by pressure [15] or fieldsweep [16]. Due to the weak magnetic anisotropy of the CeIn cubic latticeby contrast to the large anisotropy of the CeRu Si tetragonal lattice, H c collapses at p c in CeIn [17].Thus the intercept of the crossover line H M with the critical AF line H c depends on specific conditions like the Fermi surface topology, the Isingor Heisenberg character of the local magnetization, the anisotropy of theintersite interactions, the AF wavevector, or the interplay between the AFand the Kondo fluctuations. For CeRu Si , three different wavevectors k =(0 . , , k = (0 . , . ,
0) and k = (0 , , .
35) are hot spots [18]. ForLa doping antiferromagnetism develops in a transverse mode at the orderingvector k while Rh doping series order at k in a longitudinal mode. Forboth series the sublattice magnetization is aligned along the c axis. Thisdifference in the magnetic ordering is associated in the La series by the glueof H M to H c while in the Rh series a complete decoupling between H M and H c has been observed [9]. In the Rh-doped series, H c seems to collapsewith T N . When H M overpasses H c whatever is p , H M is mainly associatedwith the strength of the Kondo magnetic field H K = k B T K /gµ B required toquench significantly the Kondo effect. Below, a focus is made on the situationwhere H M touches H c below p c at finite temperature. In this case, H M is thecombined result of short range intersite correlations and local fluctuations.Basically, it is the extension above p c of the magnetic critical end point H ∗ c .From the temperature dependence of the specific heat (Fig. 4) and alsofrom inelastic neutron scattering response [19,10,20], CeRu Si appears tobe well described by the spin-fluctuation theory of Hertz, Millis, and Moriya(HMM) [21,22,23], which was first developed and tested to describe of weakantiferro- or ferromagnetism in 3 d intermetallics [24]. For HFCs a novelty isthe necessity to use an already renormalized Fermi temperature T F whichis crudely associated with the Kondo temperature of the 4 f ions [22]. Thisscheme seems now well established by the recent collection of inelastic neu-tron scattering data on the Ce − x La x Ru Si series for different La concen-trations ranging from x = 0 . x = 0, i.e. covering the sweep from the AFto the PM ground states [10]. As shown in Fig. 5, the real part of the suscep-tibility χ ′ ( Q , T ), measured at the momentum transfer Q characteristic ofthe AF order parameter (at the wavevector k ), has a maximum value at thecritical temperature T N , which collapses at the critical concentration x c . Op-positely, no maximum can be detected at a momentum transfer Q far fromthe AF hot spots. Basically, the temperature range where χ ( Q ) saturates aswell as the amplitude of χ ( Q ) is governed by the volume dependence of T K .This study has shown that fluctuations of the AF order parameter govern thetransition from paramagnetism to antiferromagnetism in Ce − x La x Ru Si .A key hypothesis in the spin-fluctuation approach is the invariance of theFermi surface through p c . Fortunately due to the high quality of the crystals,CeRu Si is one of the rare cases of HFC where the Fermi surface has beenfully determined [25,26,27,28] and which allows a serious comparison withband structure calculations. The topology of the Fermi surface appears well described in a model where the 4 f electrons are treated as itinerant andthe crystal field is taken into account [29]. Due to the improvements in theaccuracy of angular resolved photoemission spectroscopy (ARPES) it wasrecently demonstrated that this PM topology of the Fermi surface is mainlypreserved in the AF situation of CeRu (Si . Ge . ) (see Fig. 6) in sharpcontrast with a 4 f localized treatment where the Fermi surface should looklike that of LaRu Si [30]. Experiment and theory [31,32] support an itinerantpicture of the 4 f electron whatever is the ground state (AF, PM or PPM).Of course, at the magnetic ordering, a Fermi surface reconstruction might begoverned by the new AF Brillouin zone.To summarize these studies with chemical doping, which can be consid-ered as equivalent to studies under pressure, macroscopic and also micro-scopic measurements seem to support strongly a spin-fluctuation approach.However, from the temperature variation of the specific heat divided by thetemperature shown in Fig. 4, a conventional behavior of the AF fluctuationswith a singularity of the extrapolated value γ = ( C/T ) T → K in √ p − p c or √ x − x c does not seem to be reproduced. For example, at a La concentration x = 0 .
13 (which corresponds to a pressure of p c − . C/T shows asharp jump at T N ≈ C/T remains almost constant near a critical value γ c ≈
600 mJ mole − K − , which is exactly the low temperature value of C/T at the critical con-centration x = x c = 0 .
075 [7,20]. For x = 0 .
13, the transition at T L ≈ .
7K corresponds to a change in the magnetic structure which opens out ina decrease of
C/T at lower temperatures. This phenomena is very clear at x = 0 .
2. Surprisingly, focus on AF quantum criticality has been made mainlyon the PM side ( p = p c + ǫ ) and attempts to describe the AF side have beencompletely omitted. Another negative point for the HMM approach is thatthe usual link between the dependence of T N with the size of the sublatticemagnetization ( T N ∼ M / ) has not been verified [1,33]. Even on the PMside, a tiny ordered moment M ≈ . µ B survives in CeRu Si [34]. Thisresidual antiferromagnetism is up to now believed to originate from latticeimperfections which may create locally pressure gradients of a few kbar andthus would play the role of nucleation centers for the occurrence of resid-ual AF droplets. Finally, to our knowledge, no divergence of the magneticcorrelation length at p c or x c has been reported from the experiment forany heavy fermion compound [1]. So the definitive proof of a second orderquantum criticality is missing. A clear signature is that γ reaches a criticalvalue at p c corresponding to a critical value of the average effective mass m ⋆c = m ⋆ ( p c ).The interplay of the different mechanisms involved in the field restora-tion of the PM phase from an AF ground state is well shown in Fig. 7awhere the Ce − x La x Ru Si antiferromagnet of concentration x = 0 . ∂M/∂H of the magnetizationis shown as a function of H at different temperatures [7]. On cooling be-low T N ∼ H a and H c emerges clearly. However, just above H c a shallow maximum persists at H M . On warming the differentiation between H c and H M increases. Above T N only the broad maximum at H M persists. H c is characteristic of the spin flip of the static magnetization [35] while H M is governed by the inter-play of FM, AF, and local spin dynamics [36,37]. For the PM ground state,only a pseudo-metamagnetic transition survives; furthermore its characteris-tic magnetic field H M corresponds to a critical value M c = 0 . µ B /Ce-ion ofthe magnetization M , i.e. of the magnetic polarization. It is remarkable thatunder pressure, even at 1 GPa above p c , the pseudo-metamagnetic crossovercorresponds to M ( H = H M ) = M c [38]. As for H ∼ H M , the strength of theinelastic electronic scattering is mainly pressure invariant, so that m ⋆ ( H c )for p < p c might be nearly equal to m ⋆ ( H M ) for p > p c .The pseudo-metamagnetic transition in CeRu Si has been highly stud-ied by magnetization [1,39,40,41], specific heat [7,42,43,44], transport [19,45], ultrasound [46], NMR [47], elastic and inelastic neutron measurements[1,48,49], as well as quantum oscillation methods [25,26,27]. It is worthwhileto remark that at H = 0, AF correlations have been detected by neutronscattering up to T corr ≈
60 K [48] pointing out that their onset occurs farabove the Kondo temperature T K ∼
20 K, which is assigned by simple con-siderations on the specific heat maximum. Under magnetic field, a sensitivetool to detect the field of the metamagnetic transition H M ( T ) has been thedetermination of the maximum of the magnetoresistance at constant tem-perature [39]. It has been shown that H M collapses also around T = 60 Kand that it reaches a constant value only below 1 K. This is shown in Fig.8. Thermal expansion measurements at different magnetic fields were a quitepowerful tool to draw the crossover boundaries in Fig. 9 [40]: below H M , the( H, T ) boundary looks like that of an AF state, but it consists in a param-agnetic phase with strong antiferromagnetic fluctuations, and above H M itlooks like that of a ferromagnet, but is then a crossover to a PPM state.The demonstration that FM fluctuations play a major role in the sharp-ness of the pseudo-metamagnetic phenomena has been obtained via two in-elastic neutron experiments [36,37] with the observation that close to struc-tural Bragg reflections at Q = (0 . , ,
0) a strong field-induced softening ofthe FM fluctuations occurs on approaching H M (see Fig. 10). In contrast,the vanishing of the AF fluctuations under magnetic field does not occur viaan AF instability but via an increase of the damping of the AF fluctuationswith field. Macroscopically, the consequence of the switch under field fromdominant AF to FM interactions is the increase of the Sommerfeld coefficient γ = C/T for T → H M is quite analogous to that predicted by the AF spin-fluctuation theory. However, as we stressed above the real mechanism whichdrives the transition is a transfer from AF to FM fluctuations. From thesestudies, a key message is that at H M , γ ( H M ) has almost the same value thanthat reached at p c so that m ⋆ ( p c ) = m ⋆ ( H M ).Such a convergence of the effective mass under pressure and magneticfield was recently reported for the antiferromagnet CeRh Si at p = 0 (Fig.12) [12]. Comparing the field and pressure variation of the A coefficient ofthe T Fermi-liquid term of the resistivity (which is assumed to be propor-tional to γ ), led to the conclusion that A ( p c ) ∼ A ( H c ) (see Fig. 13), i.e. m ⋆ ( p c ) ∼ m ⋆ ( H c ) [12]. Furthermore we stress that a marked field enhance-ment of A starts even far below H c , at a field near H ⋆ ∼
15 T where at T = 0 the crossover between the PM and PPM phases would occur in absence of an-tiferromagnetism. Thus, an important observation is that despite the strongfirst order nature of the metamagnetic transition of CeRh Si at H c (wherethe magnetization jumps by ∆M ∼ . µ B /Ce-ion), the field enhancementoccurs far below H c , i.e. at ( H ⋆ − H c ) /H c ∼ .
5. In Fig. 14 we have drawnschematically the (
H, p, T ) phase diagram of CeRh Si which is also gen-eral for other HFC. The dashed area indicates the region in the ( p, H ) planewhere FM fluctuations might play a significant role. Far above p v , wherevalence fluctuations are expected to be large, the FM fluctuations shouldcertainly drop. However, as will be discussed later, the FM fluctuations maybe enhanced near p v , even at H → A ( p c ) and A ( H c )can be found in the study realized on CeIn under pressure and magneticfield and on CeIn . Sn . , where Sn-doping permits to lower H c down to45 T instead of 60 T for the pure compound [16,17]. If we consider thata broadening is induced by doping, the data are consistent with m ⋆ ( p c ) ≈ m ⋆ ( H c ) (Fig. 15). Another similarity between the three discussed examplesof CeRu Si , CeRh Si and CeIn is that a large magnetic polarization isrequired to destroy AF correlations at the profit of FM correlations. Onemay hope to detect fully the Fermi surface in the AF and PPM phases byquantum oscillations techniques and even later to zoom on the Fermi surfaceevolution through the sharp crossover regime at H M . However, very often inthe experiments large parts of the Fermi surface have not been observed asfor example minority spin carriers may get a too large effective mass to bedetected [1]. For example in CeRu Si , the field enhancement of m ⋆ above H M is only observed for a few orbits and its correspondence with the averagevalue measured via the γ term cannot be verified [26,27].Emerging from macroscopic measurements on the quite different systemsCeRu Si , CeRh Si , and CeIn , a golden rule has to be obeyed in order tofind the relation m ⋆ ( p c ) = m ⋆ ( H M or H c ). Furthermore this equality does notrequire to be at the AF quantum singularity at p c . For conventional magneticmaterials, it is well known that applying a magnetic field generally changesthe universality class of the phase transition. Thus the similarity betweenpressure and magnetic field tunings is not obvious. As pointed out, close to p c , the AF phase has been weakly considered and discussed. A sound idea isto look more carefully to the microscopic phenomena. The independence ofthe product Γ q χ q of the magnetic relaxation rate Γ q and the susceptibilityat the wavevectors q derived in the framework of a quasi-localized model [50]seems also obeyed in HFCs, whatever is the ground state [10]. The magneticfield induced transfer from AF to FM fluctuations may be dominated by sucha rule taking into account that the first order nature of the FM instability, aswell as the damping of FM fluctuations under field, prevent any collapse of Γ q at q = 0 for H = H M . Thus, there is a concomitant mechanism to avoidany divergence of m ⋆ at p c and at H c . Finding a H - or p -induced singularityin the density of states to explain the common convergence of m ⋆ ( p c ) with m ⋆ ( H M ) is a key issue.Of course, an appealing route is to look deeper than before to the Fermisurface instability with the idea that the singularity has to be marked in the p and H evolution of the Fermi surface. Such instabilities may not require adrastic change but a topological change as discussed long time ago for the 2.5Lifshitz transition [51]. In CeIn this approach was already made to explainthe field-induced evolution of the Fermi surface [52]. It was demonstratedthat, via magnetic polarization, the magnetic field can lead to a logarithmicdivergence of the observed de Haas van Alphen mass; the key point is thefield evolution of the topological change which has occurred at the onset ofantiferromagnetism via the modification in the balance between majority andminority spin carriers and the spin dependences of the effective mass of theelectrons. Quantum and topological criticalities of a Lifshitz transition havebeen discussed for two-dimensional correlated electron systems [53]. Up tonow, no similar study exists for the case of a three-dimensional system.To explain the pseudo-metamagnetism in CeRu Si , a pseudo-gap modelwas introduced as an input parameter in the periodic Anderson model [54].The field sweep in the pseudo-gap induces a change in sign of the exchange atthe metamagnetic crossover field H M from an AF to a FM exchange. Quiterecently, a phenomenological spin-fluctuation theory for an AF quantum-tricritical point (QTCP) has been developed [55]. This model is quite suit-able for HFC like CeRu Si and CeRh Si which present both metamagneticphenomena. Around the QTCP, both critical AF fluctuations (at Q ) and FMfluctuations play an equivalent role in the mass enhancement. The particu-larity is that the singular dependence of γ is equal to that of a conventionalAF quantum critical point. In this model no power law divergence of thespecific heat in three dimensions is predicted, and at the critical field H c thesingular Sommerfeld coefficient γ ( H c ) is finite and given by equal footing bythe AF and magnetic field-induced FM fluctuations. This is in agreementwith our experimental observations reported here. The theoretical discus-sion takes only the singular part of the Sommerfeld coefficient γ Q into ac-count. The renormalized bands lead to an additional normal contribution γ B through quasi-local fluctuations which is quite comparable to γ Q . Thedifference between γ ( p c ) and γ ( H c ) is reduced to a factor 1.5. Furthermore, γ B itself is linked to the Kondo effect (i.e. to the Kondo field H K ) and willdecrease with magnetic field monotonously. This will push γ ( H c ) quite closeto γ ( p c ). So, a reduced maximum in the field dependence of γ at H c can beexpected. From the field variation of C/T at the verge to antiferromagnetism(e.g. for a concentration x = 0 . − x La x Ru Si in vicinity of the x c )no clear maximum of γ ( H ) is observed. However, the correction by γ B ( x c )may restore a maximum. Up to now a careful inelastic neutron scatteringstudy under magnetic field for the CeRu Si series has been performed onlyabove p c to understand the pseudo-metamagnetic phenomena. No divergenceof χ Q was observed at H M but only a strong damping of the AF correlations[56]. According to the theory, χ − Q will be strongly reduced far from H ⋆c ,( χ − Q ∼ | H − H ⋆c | ) while the uniform susceptibility is predicted to have onlya p | H − H ⋆c | dependence. Thus a new set of experiments with tuning thefield through H c and H M just below p c on the AF side of the quantum phasetransition is necessary.For the reported Ce cases, the FM fluctuations are induced by the mag-netic field. In Yb-based HFC the interplay between valence and magnetic transitions is certainly strong due to the weakness of the hybridization [57].FM interactions have been observed to be enhanced when both valence andmagnetic fluctuations interact [58]. The physical argument is that large dy-namical volume fluctuations, which involve a q → Si [59,60],one may expect a drastic change under magnetic field even for the low energymagnetic excitations [57,58]. Our own view is [60] that even in YbRh Si nodivergence of the effective mass occurs at H c (which is not a metamagnetictransition in the case of YbRh Si ). We did not observe a divergence of the A coefficient of the T term of the resistivity and further, the upper temper-ature T A of the T law remains finite at H c . One particularity of YbRh Si is that γ Q /γ B is large.The next issue appears for us to find a material where a full determinationof the Fermi surface would be possible in each phase, since actually for thethree different cases considered here, the observation of large parts of theFermi surface is still missing, notably in the PPM phase. For the other casesof highly studied HFC with initially huge values of the effective mass ( γ ∼ − K − ) close to quantum singularity as CeCu [61] or YbRh Si [62],the low electronic mean free path of the first and the low value of H c for thesecond make very unlikely the opportunity of a direct measurement of theFermi surface. Thus, the stimulating challenge is clearly to observe completelythe Fermi Surface of the different AF, PM, and PPM phases.JF thanks Pr M. Imada, Y. Kuramoto, and K. Miyake for theoreticaldiscussions. This work was performed through the support of the ANR Deliceand of Euro- magnet II via the EU contract RII3-CT-2004-506239, and thestay of JF in Osaka through the global excellence network. References
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Fig. 2
Temperature dependence of the Gr¨uneisen parameter of CeRu Si (afterRef. [5]).2 H pp c H c H M ~ H K H K H c CeIn CeRu Si H* Fig. 3
Pressure dependence of the critical fields H c and H M for the case ofCeRu Si and CeIn [1]. Ce La x Ru Si x = 0x = 0.075x = 0.13x = 0.2 C m / T ( J / m o l K ) T (K) T L T N T N T L Fig. 4
Temperature dependence of magnetic contribution to the specific heat C m divided by temperature T of Ce − x La x Ru Si for x = 0 (which corresponds to apressure p c + 0 . x = 0 .
075 ( p c ), x = 0 .
13 ( p c − . x = 0 . p c − . x = Q = ( c ' ( Q , T ) ( a r b . un it ) C
13 % Ce La Ru Si x x ) x (a) T (K) Q = (0.44,1,0) x = 20 %
13 % c ' ( Q , T ) ( a r b . un it ) x ) C (b) T (K) Fig. 5 (a) Temperature dependence of the real part of the static susceptibility χ ′ ( Q ) at the wave vector of the AF fluctuations at Q and (b) at a wave vector Q = Q very far from the AF wavevector [10].4 (d) (e) XXZ Γ CeRu Si LaRu Si Fig. 6 (a-c) ARPES results on CeRu Si and CeRu (Si . Ge . ) compared withLaRu Si . (d,e) Fermi surface calculated for CeRu Si and LaRu Si . (after Ref.[30]) a b Fig. 7 a) Differential susceptibility of Ce . La . Ru Si at different temperaturesas a function of the applied field. b) Temperature variation of the critical metam-agnetic field H a and H c and of the pseudo-metamagnetic field H M [7].5 H ( T ) T (K) HH M CeRu Si Fig. 8
Temperature dependence of H M for CeRu Si . H M is derived from themaximum of the positive magnetoresistance measured at constant temperature. ∆H gives the broadening of the maximum. (see Ref. [39])) x = 0 x = 0.05 T M ( H ) / T M ( ) H / H M N - "AF" PPMPMCe x La x Ru Si Fig. 9
Crossover boundaries of the different regimes of the PM phase ofCe − x La x Ru Si with at low field the nearly AF domain (N-”AF”) and at highfield the polarized paramagnetic phase PPM. [1,39,40]6 ( a r b . un it s ) H (T) b) CeRu Si Q = (0.9, 1, 1)T = 0.4 K n e u t r on i n t e n s it y ( a r b . un it s ) H (T) a)CeRu Si Q = (0.9, 1, 0)E = 0.4 meVT = 170 mK
Fig. 10
Evidence of the softening of the FM fluctuations at H M for Q = (0 . , , T = 0 . E = 0 . Γ detected in the inelastic spectrum and field variation of χ (0) which is in differenceto the uniform susceptibility strongly enhanced by the huge magnetostriction at H M (taken from Ref. [37]).7 g ( J / K m o l ) H (T) x = x c = 0.075 [27] x = 0 [31] x = 0 [35] x = 0 [33] x = 0.05 [31] x = 0.1 [7] Ce La x Ru Si Fig. 11
Field variation of γ = C/T of CeRu Si and Ce − x La x Ru Si . γ ( p c ) ismeasured at x = x c at p = 0. For x = 0 see Refs. [40,42,43], for x = 0 .
05 Ref. [40],for x = 0 . x = x c = 0 .
075 Ref. [20]. = thermal expansion )( , = resistivity ; , = torque ; H M T ( K ) T N H (T) H c PPMPMAF H * CeRh Si Fig. 12 ( H, T ) phase diagram of CeRh Si at p = 0. For clarity we have withdrawnthe different phases which occur in the AF phase. The PPM boundary in absenseAF order seems to end up at H ⋆ ∼
15 T for T → H c (0) = 27 T. A ( H ) is strongy enhanced above H ⋆ .8 -1.0 -0.5 0.0 0.5 1.00.00.51.0 - c )/ c A / A m a x CeRh Si = H || c = p Fig. 13
Enhancement of the A coefficient of the resistivity under pressure com-pared to the enhancement under magnetic field on a normalized field scale forCeRh Si [12]. H H c Polarized regimeAF order Fermi liquid p c pT T*T N T K D CF H c* p v H M CeRu Si ( p = 0) CeRh Si ( p = 0) Fig. 14 ( H, p, T ) phase diagram of CeRh Si . The dashed area correspond to thedomain where the FM component will play a major role in the enhancement of γ .9 T ( K ) (H - H c ) / H c CeIn Sn AF A ( c m / K ) -1.0 -0.5 0.0 0.5 1.00510 CeIn (p - p c ) / p c T ( K ) AF A ( c m / K ) Fig. 15 a) (
T, H ) phase diagram of CeIn . Sn . . The doping is used to lower H c from 60 T for the pure compound CeIn to 42 T with Sn doping. This allows todetermine A ( H ) in the vicinity of H c [16]. b) ( p, T ) phase diagram of CeIn3