We report a comprehensive study of CeIn$_{3-x}$Sn$_x$ $(0.55 \leq x \leq
0.8)$ single crystals close to the antiferromagnetic (AF) quantum critical
point (QCP) at $x_c\approx 0.67$ by means of the low-temperature thermal
expansion and Gr\"uneisen parameter. This system represents the first example
for a {\it cubic} heavy fermion (HF) in which $T_{\rm N}$ can be suppressed
{\it continuously} down to T=0. A characteristic sign change of the Gr\"uneisen
parameter between the AF and paramagnetic state indicates the accumulation of
entropy close to the QCP. The observed quantum critical behavior is compatible
with the predictions of the itinerant theory for three-dimensional critical
spinfluctuations. This has important implications for the role of the
dimensionality in HF QCPs.Comment: Physical Review Letters, to be publishe

Caroca-Canales, N.

Custers, J.

Gegenwart, P.

Geibel, C.

Kuechler, R.

Sereni, J. G.

Steglich, F.

Stockert, O.

English

arXiv

Küchler, R.

Sereni, J.

MPG.PuRe

Quantum Criticality in the Cubic Heavy-Fermion System CeIn<sub>3-x</sub>Sn<sub>x</sub>

Max Planck Society eDoc Server

Quantum Criticality in the Cubic Heavy-Fermion System CeIn3-xSnx

Gegenwart, Philipp

OPUS Augsburg

Quantum Criticality in the Cubic Heavy-Fermion System CeIn3xSnxR. Küchler,1 P. Gegenwart,1 J. Custers,1 O. Stockert,1 N. Caroca-Canales,1 C. Geibel,1 J. G. Sereni,2 and F. Steglich11Max-Planck Institute for Chemical Physics of Solids, D-01187 Dresden, Germany2Laboratorio Bajas Temperaturas, Centro Atomico Bariloche, 8400 S.C. Bariloche, Argentina(Received 1 February 2006; published 30 June 2006)We report a comprehensive study of CeIn3xSnx (0:55  x  0:8) single crystals close to theantiferromagnetic quantum-critical point (QCP) at xc  0:67 by means of the low-temperature thermalexpansion and Grüneisen parameter. This system represents the first example for a cubic heavy fermion inwhich TN can be suppressed continuously down to T  0. A characteristic sign change of the Grüneisenparameter between the antiferromagnetic and paramagnetic states indicates the accumulation of entropyclose to the QCP. The observed quantum-critical behavior is compatible with the predictions of theitinerant theory for three-dimensional critical spin fluctuations. This has important implications for therole of the dimensionality in heavy-fermion QCPs.DOI: 10.1103/PhysRevLett.96.256403 PACS numbers: 71.27.+a, 71.10.HfNon-Fermi-liquid (NFL) properties are observed inmany heavy-fermion (HF) systems and frequently attrib-uted to a nearby quantum-critical point (QCP) [1]. A QCPcan arise by continuously suppressing the transition tem-perature TN of an antiferromagnetic (AF) phase to zero,e.g., by chemical or applied pressure or an external mag-netic field. QCPs are of great current interest due to theirsingular ability to influence the finite temperature proper-ties of materials. Heavy-fermion metals have played thekey role in the study of AF QCPs. The essential question ishow the heavy quasiparticles evolve if these materials aretuned from the paramagnetic into the AF ordered state. Thetraditional picture describes a spin-density-wave (SDW)transition and related, a mean-field type of quantum-critical behavior. Here the quasiparticles retain their itin-erant character [2,3]. Unconventional quantum criticality,which qualitatively differs from the standard theory of theT  0 SDW transition, may arise due to a destruction ofKondo screening. Here the quasiparticles break up intotheir components: conduction electrons and local 4f mo-ments forming magnetic order [4,5]. This locally criticalpicture leads to a number of distinct properties, includingstronger than logarithmic mass divergence, !=T scaling inthe dynamical susceptibility, and a large reconstruction ofthe Fermi surface. Such behavior has been found at least insome HF systems [6–8]. The central question is to identifythe crucial parameter leading to the different types ofQCPs. Of particular importance should be the dimension-ality of the magnetic fluctuations, which could be reducedby the presence of frustration. It is proposed in Ref. [4] thatfor magnetically three-dimensional (3D) systems withoutfrustration the itinerant SDW picture should apply. On theother hand, 2D magnetic systems should be described by alocally quantum-critical picture [4]. However, systemscurrently under investigation are either tetragonal, e.g.,CeNi2Ge2, YbRh2Si1xGex2 [9], and CeCu2Si2 [10],hexagonal, e.g., YbAgGe [11], or monoclinic, e.g.,CeCu6xAux [6], and the lower crystallographic symmetrycould result in fluctuations with reduced dimensionality.Therefore, the dimensionality of the critical spin fluctua-tions clearly needs to be substantiated by inelastic neutronscattering experiments. In order to avoid this constraint,experiments on cubic systems close to QCPs are particularinteresting. CeIn3xSnx, with a cubic point symmetry of Ceatoms in the Cu3Au structure (compare the inset in Fig. 1),is thus an excellent candidate for such a study, as here low-dimensional spin fluctuations can be ruled out. Thus, theinteresting question arises whether the mechanism of NFLbehavior in this system can be described by an itinerant3D-SDW theory.In this Letter, we present thermal-expansion measure-ments and a Grüneisen ratio analysis performed on single-crystalline samples of the cubic system CeIn3xSnx closeto the critical concentration xc  0:67, where TN is sup-pressed to zero by doping. Recently, it has been shown thatthe thermal volume expansion   V1dV=dT (V: sam-ple volume) is particularly suited to probe quantum-criticalbehavior, since, compared to the specific heat, it is muchmore singular in the approach to the QCP [12]. As aconsequence, the Grüneisen ratio  =C of thermalexpansion T to specific heat CT is divergent as Tgoes to zero at any pressure-sensitive QCP, and the asso-ciated critical exponent can be used to distinguish betweenthe different types of QCPs. In the itinerant scenario, thedivergence  / 1=T is given by   1=z [12], with  thecritical exponent for the correlation length  / jrj (r:distance from the QCP) and z the dynamical critical ex-ponent in the divergence of the correlation time c / z.For a 3D AF QCP,   1=2 and z  2, yielding   1.Thus, a study of the Grüneisen ratio can prove the validityof the 3D-SDW picture in the title system provided that  1.The magnetic x; T phase diagram of polycrystallineCeIn3xSnx has been widely studied for 0  x  1 bysusceptibility [13], specific-heat [14], and resistivity mea-surements [15,16] (see Fig. 1). Whereas TN for undopedPRL 96, 256403 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending30 JUNE 20060031-9007=06=96(25)=256403(4) 256403-1 © 2006 The American Physical SocietyCeIn3 vanishes discontinuously below 3 K under hydro-static pressure [17], it can be traced down to 0.1 K forCeIn3xSnx, and an additional first-order phase transitionTI has been found for 0:25< x< 0:5 inside the AF state[14]. These differences are related to the change of theelectronic structure induced by Sn doping. Beyond a pos-sible tetracritical point at x  0:4 [14], an almost lineardependence of TNx is observed. This is in contrast toTN / xc  x2=3 predicted by the 3D-SDW theory [2].Thus, the origin of the NFL behavior in this system re-mains an open question, and further thermodynamic stud-ies are needed to shed light on the nature of the QCP.The CeIn3xSnx single crystals investigated here(0:55  x  0:80) were grown by a Bridgman-type tech-nique. Large single crystals with a mass of 15 g wereproduced, analyzed by x-ray powder diffraction, and foundto be of single phase with the proper cubic structure.Within the 2% accuracy of the x-ray diffraction, noimpurity phases were resolvable. Thin bars with a length2  l  6 mm, suitable for the dilatometric investigations,were cut out. The thermal expansion has been measured ina dilution refrigerator using an ultrahigh resolution capaci-tive dilatometer with a maximum sensitivity correspondingto l=l  1011.Figure 2(a) shows the volume thermal expansion  ofsingle-crystalline CeIn3xSnx with x  0:55, 0.65, 0.7, and0.8 plotted as T=T vs logT. The volume-expansioncoefficient  is given by   3	 , with  being thelinear thermal-expansion coefficient. For x  0:55, thebroadened steplike decrease in =T at TN  0:6 K marksthe AF phase transition, in perfect agreement with specific-heat measurements on the same single crystal [18]. Uponincreasing the concentration, we find for x  0:65 and 0.7diverging behavior over nearly two decades in T down to80 mK. These data suggest that TN is suppressed at acritical concentration xc  0:67 0:03, also consistentwith specific-heat measurements performed on the samesamples [18]. Finally, for x  0:8 we recover Fermi-liquidbehavior T=T  const for T ! 0.In the following, we will analyze the observed NFLbehavior and make a comparison with the predictions ofthe itinerant SDW scenario [12]. A best-fit description ofthe x  0:65 data in the entire temperature range 0:08 K T  6 K according to =T  a0  a1Ta2 reveals a2 0:4 0:01 [see the dashed line in Fig. 2(a)]. However,as shown in the upper part of Fig. 2(b), the deviationbetween the data and this fit shows several broad bumps,indicating that the fit does not properly describe the data.We therefore tried best power-law fits for 0:08 K  T Tmax with varying Tmax. For Tmax  1 K, the fit is ofexcellent quality [cf. solid line in Fig. 2(a) and lowestpart of Fig. 2(b)] and the resulting exponent equals 0:5,i.e., the value predicted by the 3D-SDW scenario [12].We now turn to the Grüneisen parameter defined as  Vm=T=C, where the constants Vm and T denote themolar volume and isothermal compressibility, respectively.-5051015200.1 1 6β / T (10-6K-2)CeIn3-xSnxx = 0.65x = 0.7x = 0.8x = 0.55aT (K)000-1x = 0.65(β / T) − (a0+a 1Ta 2) (10-6 K-2)a2 = -0.5a2 = -0.45a2 = -0.4bFIG. 2. (a) Volume thermal-expansion coefficient  ofCeIn3xSnx single crystals as =T vs logT. The gray solid andblack dashed lines indicate T0:5 and T0:4 dependencies, re-spectively. The arrow indicates AF phase transition.(b) Deviation of =T data for a x  0:65 sample from bestpower-law fits for T  1 K (squares), T  2 K (circles), andT  6 K (triangles), respectively, as =T  a0  a1Ta2  vslogT. For clarity, the three data sets have been shifted bydifferent amounts vertically.0 0.5 10510TI LFLAFT*TNCeIn3-xSnxT(K)xFIG. 1. Magnetic phase diagram for cubic CeIn3xSnx (x 1). The solid circles and diamonds indicate TN , determined fromspecific-heat [14] and electrical resisitivity [15] measurements,respectively. The open diamonds mark T?, the upper limit ofLandau Fermi-liquid behavior, e.g., T / T2 [16]. The opentriangles indicate first-order transition TI [14].PRL 96, 256403 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending30 JUNE 2006256403-2The specific heat has been studied in the temperature range40 mK  T  4 K on the same CeIn3xSnx samples usedfor thermal expansion [18,19]. Here the nuclear quadru-pole contribution of indium which becomes importantbelow 150 mK has been subtracted. Figure 3 shows acomparison of T for all samples studied in thermalexpansion. Since the temperature dependence of specificheat is much weaker compared to that of thermal expan-sion, its influence to the Grüneisen parameter is rathersmall. Therefore, the variation of T for the differentCeIn3xSnx samples is very similar to that found in =T(compare Fig. 2). Both single crystals closest to xc, x 0:65 and x  0:7, show a divergent behavior down to thelowest accessible temperature with very large  values at0.1 K which are of similar size as found for other quantum-critical HF systems [9,20]. On the other hand, saturationis observed for x  0:55 and x  0:8 being located in theAF ordered and the Fermi-liquid regime, respectively. Thefact that the divergence of T in the quantum-criticalregime is stronger than logarithmic (compare the double-logarithmic representation of the x  0:65 data presentedin the inset in Fig. 3) provides clear evidence for a welldefined (pressure-sensitive) QCP in the system. If thedisorder present in the system would lead to a ‘‘smeared’’quantum-critical regime, T could diverge at most loga-rithmically [12].Another indication for a QCP is the sign change of theGrüneisen parameter between the ordered and the disor-dered regime. As discussed in Ref. [21], it is directlyrelated to the entropy accumulation near the QCP. Thedifferent signs of  in the AF and paramagnetic regionsreflect the opposite pressure dependencies of the respectivecharacteristic energy scales. Below TN , the effective AFintersite interaction dominates, whose negative pressuredependence gives rise to a negative Grüneisen parameter< 0. On the other hand, the positive Grüneisen ratio inthe paramagnetic state is compatible with the positivepressure dependence of the Kondo temperature in Ce-based HF systems.In order to compare our results for the x  0:65 samplewhich is located closest to the QCP with the theoreticalpredictions for an itinerant AF QCP [12], we need tocalculate the critical Grüneisen ratio crT / cr=Ccr ofcritical contributions to thermal expansion and specificheat. For thermal expansion, crT  T  a0T, witha0  0:3	 106 K2 as determined from the best fit up to1 K; see above. Within the itinerant theory for the 3D AFcase, the critical contribution to specific heat is subleading[12]: CcrT  CT  	0T, with Ccr < 0, Ccr=T ! 0 forT ! 0, and 	0  C=TjT0. For 	0, we use the value0:851 Jmol1 K2 obtained in Ref. [18] from fitting thelow-temperature electronic specific heat in a restrictedtemperature range 0:3 K  T  1:4 K according toC=T  	01 a0Tp. Figure 3 displays a log-log plot ofcrT versus temperature. We find cr / T, with anexponent   1:1 0:1 which is very close to 1, predictedby the itinerant theory. Note that this exponent is ratherinsensitive of 	0 subtracted from the specific-heat data:Using 	0  0:9 and 0:95 Jmol1 K2 results in   1:07and 1.02, respectively. Interestingly, the exponent for thecritical Grüneisen ratio, which theoretically equals thedimension of the most relevant operator that is coupledto pressure [12], holds over a much larger temperaturerange than the respective 3D-SDW dependencies in spe-cific heat [18] and thermal expansion (cf. Fig. 2). A similarobservation has also been made for CeNi2Ge2 [9].For those two systems for which an unconventional QCPhas been proposed, YbRh2Si2 and CeCu6xMx (M  Au,Ag), distinctly different temperature dependences havebeen observed: cr / T0:7 in the former [9] and cr /logT [20] in the latter case. It is proposed in Ref. [4] thatfor magnetically 3D systems without frustration the SDWpicture should apply. This is consistent with our Grüneisenratio analysis.For the 3D AF case, the itinerant theory predicts anasymptotic T3=2 dependence for the temperature dependentpart to the electrical resistivity [2,3]. As discussed inRef. [22], the interplay between strongly anisotropic scat-tering due to the critical spin fluctuations and isotropicimpurity scattering can lead at elevated temperature totemperature exponents of the resistivity between 1 and1.5, depending on the amount of disorder. SystematicT studies down to mK temperatures on polycrystallineCeIn3xSnx revealed an almost linear temperature depen-0.1 10501000.1 1 1020100      x 0.65 0.7 0.8 0.55CeIn3-xSnxΓT (K)FIG. 3. Temperature dependence of the Grüneisen parameter  Vm=T=C of several CeIn3xSnx single crystals asT vs logT. Vm  6:25	 105 m3 mol1 and T  1:49	1011 Pa1 [26] are the molar volume and isothermal compres-sibility, respectively. The arrow indicates AF phase transition.The inset displays data for x  0:65 in a double-logarithmic plot.The dotted line indicates the power-law dependence  / T0:31.PRL 96, 256403 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending30 JUNE 2006256403-3dence in the quantum-critical regime [15]. Similar behav-ior is observed for single-crystalline CeIn2:35Sn0:65 as well;see the inset in Fig. 4. However, due to the high Sn dopingneeded to tune the system towards the QCP, the resistivityratio 300 K=0 is of the order of 1 and the temperaturevariation amounts to a few percent of 0 only, making thecomparison with theoretical predictions very difficult. Thisindicates that transport experiments alone are not sufficientto characterize quantum criticality in disordered systems.Possibly, also the slope of TNx differs from the 3D-SDWprediction because disorder is not constant but increaseswith increasing x. However, the algebraic divergence ofT for T ! 0 at x  xc proves a pressure-sensitive QCPin the system and excludes disorder-driven scenarios forthe observed NFL behavior [12].In conclusion, our study on CeIn3xSnx single crystalsby means of the low-temperature thermal expansion andGrüneisen parameter has proven the applicability of theitinerant theory for 3D critical spin fluctuations in thiscubic system. Since strong contradictions to this theoryhave been found in systems such as CeCu5:9Au0:1 [6] orYbRh2Si1xGex2 [7] with lower crystallographic sym-metry and, at least in the case of the former system,strongly anisotropic quantum-critical fluctuations, the pa-rameter dimensionality obviously plays an important rolefor the nature of HF QCPs. We tentatively classify thedifferent HF systems studied by Grüneisen analysis at theirrespective QCPs as follows: (i) CeIn3xSnx and CeNi2Ge2[9], for which the latter system neutron scattering mea-surements revealed 3D low-energy magnetic fluctuations[23], show thermodynamic behavior compatible with the3D itinerant theory, whereas for (ii) YbRh2Si1xGex2 [9]and CeCu5:8Ag0:2 [20] strong contradictions to this model(for both 2D and 3D critical spin fluctuations) are ob-served. Neutron scattering has proven 2D quantum-criticalfluctuations in CeCu5:9Au0:1 [24], while for YbRh2Si2 acomplicated behavior with competing AF and ferromag-netic quantum-critical fluctuations has been observed [25].The comparison with our results on CeIn3xSnx suggeststhat a destruction of Kondo screening causing unconven-tional quantum criticality is prevented in magnetically 3Dsystems.We thank T. Radu for providing low-temperaturespecific-heat data. This work is supported in part by theF. Antorchas–DAAD cooperation program (ProgramNo. 1428/7).[1] See, e.g., S. Sachdev, Quantum Phase Transitions(Cambridge University Press, Cambridge, England, 1999).[2] A. J. Millis, Phys. Rev. B 48, 7183 (1993).[3] T. Moriya and T. Takimoto, J. Phys. Soc. Jpn. 64, 960(1995).[4] Q. Si et al., Nature (London) 413, 804 (2001).[5] P. Coleman and C. Pépin, Physica (Amsterdam) 312B–313B, 383 (2002).[6] A. Schröder et al., Nature (London) 407, 351 (2000).[7] J. Custers et al., Nature (London) 424, 524 (2003).[8] S. Paschen et al., Nature (London) 432, 881 (2004).[9] R. Küchler et al., Phys. Rev. Lett. 91, 066405 (2003).[10] P. Gegenwart et al., Phys. Rev. Lett. 81, 1501 (1998).[11] S. L. Bud’ko, E. Morosan, and P. C. Canfield, Phys. Rev. B69, 014415 (2004).[12] L. Zhu et al., Phys. Rev. Lett. 91, 066404 (2003).[13] J. Lawrence, Phys. Rev. B 20, 3770 (1979).[14] P. Pedrazzini et al., Eur. Phys. J. B 38, 445 (2004).[15] J. Custers et al., Acta Phys. Pol. B 34, 379 (2003).[16] J. Custers, Ph.D. thesis, Technische Universität Dresden,2003 (unpublished).[17] N. D. Mathur et al., Nature (London) 394, 39 (1998).[18] T. Rus et al., Physica (Amsterdam) 359B–361B, 62(2005).[19] T. Radu (née Rus), Ph.D. thesis, Technische UniversitätDresden, 2005 (unpublished).[20] R. Küchler et al., Phys. Rev. Lett. 93, 096402 (2004).[21] M. Garst and A. Rosch, Phys. Rev. B 72, 205129 (2005).[22] A. Rosch, Phys. Rev. Lett. 82, 4280 (1999).[23] H. Kadowaki et al., Phys. Rev. B 68, 140402(R) (2003).[24] O. Stockert et al., Phys. Rev. Lett. 80, 5627 (1998).[25] K. Ishida et al., Phys. Rev. Lett. 89, 107202 (2002).[26] I. Vedel et al., J. Phys. F 17, 849 (1987).0.1 1 101101000.0 0.50.850.89-ΓcrT (K)CeIn2.35Sn0.65ε = 1.1 ± 0.1ρ/ρ 300 KT (K)FIG. 4. Critical Grüneisen ratio cr  Vm=Tcr=Ccr forCeIn2:35Sn0:65 as logcr vs logT with critical components cr T  a0T and Ccr  CT  	T derived after subtraction ofbackground contributions (see text). The solid line representscr / 1=T, with   1:1 0:1. The inset shows the low-temperature electrical resistivity of a single crystal of similarcomposition.PRL 96, 256403 (2006) P H Y S I C A L R E V I E W L E T T E R Sweek ending30 JUNE 2006256403-4

Quantum criticality in the cubic heavy-fermion system CeIn3−xSnx

Quantum criticality in the cubic heavy-fermion system CeIn_{3-x}Sn_x

Quantum criticality in the cubic heavy-fermion system CeIn_{3-x}Sn_x

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