Non-Fermi liquid (NFL) behavior in the normal state of the heavy-fermion
superconductor UBe13 is studied by means of low-temperature measurements of the
specific heat, C, and electrical resistivity, \rho, on a high-quality single
crystal in magnetic fields up to 15.5 T. At B=0, unconventional
superconductivity forms at Tc=0.9 K out of an incoherent state, characterized
by a large and strongly temperature dependent \rho(T). In the magnetic field
interval 4 T \leq B \leq 10 T, \rho(T) follows a T^3/2 behavior for Tc(B)\leq T
\leq 1 K, while \rho is proportional to T at higher temperatures. Corresponding
Non-Fermi liquid behavior is observed in C/T as well and hints at a nearby
antiferromagnetic (AF) quantum critical point (QCP) covered by the
superconducting state. We speculate that the suppression of short-range AF
correlations observed by thermal expansion and specific heat measurements below
T_L \simeq 0.7 K (B=0) yields a field-induced QCP, T_L \to 0, at B=4.5 T.Comment: Presented at the M2S-2003 conference in Rio / Brazi

Bednorz

C. Langhammer

Cooper

F. Steglich

Felten

G.R. Stewart

Gegenwart

Hegger

J.S. Kim

Knetsch

Kromer

Lang

M. Lang

Mathur

Millis

Moriya

N. Oeschler

P. Gegenwart

Paglione

Petrovic

R. Helfrich

Ramirez

Rosch

Steglich

English

arXiv

Gegenwart, P.

Langhammer, C.

Helfrich, R.

Oeschler, N.

Lang, M.

Kim, J.

Stewart, G.

Steglich, F.

MPG.PuRe

Non-Fermi liquid normal state of the heavy-fermion superconductor UBe<sub>13</sub>

Kim, J. S.

Stewart, G. R.

Max Planck Society eDoc Server

Non-Fermi liquid normal state of the heavy-fermion superconductor UBe13

Gegenwart, Philipp

OPUS Augsburg

                                                         Non-Fermi liquid normal state of theheavy-fermion superconductor UBe13P. Gegenwart a,*, C. Langhammer a, R. Helfrich a, N. Oeschler a, M. Lang b,J.S. Kim c, G.R. Stewart c, F. Steglich aa Max-Planck Institute for Chemical Physics of Solids, Nothnitzer Str. 48, D-01187 Dresden, Germanyb Institute of Physics, University of Frankfurt/Main, D-60054 Frankfurt, Germanyc Department of Physics, University of Florida, Gainesville, FL 32611, USAAbstractNon-Fermi liquid (NFL) behavior in the normal state of the heavy-fermion superconductor UBe13 is studied bymeans of low-temperature measurements of the specific heat, C, and electrical resistivity, q, on a high-quality singlecrystal in magnetic fields up to 15.5 T. At B ¼ 0, unconventional superconductivity forms at Tc ¼ 0:9 K out of anincoherent state, characterized by a large and strongly temperature dependent qðT Þ. In the magnetic field interval4 T6B6 10 T, qðT Þ follows a T 3=2 behavior for TcðBÞ6 T 6 1 K, while q is proportional to T at higher temperatures.Corresponding non-Fermi liquid behavior is observed in C=T as well and hints at a nearby antiferromagnetic (AF)quantum critical point (QCP) covered by the superconducting state. We speculate that the suppression of short-rangeAF correlations observed by thermal expansion and specific heat measurements below TL  0:7 K (B ¼ 0) yields a field-induced QCP, TL ! 0, at B ¼ 4:5 T.                                   Keywords: Non-Fermi liquid behavior; Heavy-fermion superconductivity; UBe131. IntroductionUnconventional superconductivity has been thesubject of intense research in recent decades. Of partic-ular interest are those systems in which superconduc-tivity develops out of a normal state that cannot bedescribed within the Landau Fermi liquid model. Mostprominent examples include the high-Tc cuprates [1] andthe heavy-fermion (HF) superconductors [2]. For someclean Ce-based HF systems which are antiferromagnetsat ambient pressure, HF superconductivity has beendiscovered in a small pressure range around the quan-tum-critical point (QCP), where TN ! 0, to develop outof a non-Fermi liquid (NFL) state, characterized by anunusual resistivity behavior q  q0 / T e with 16 6 1:5* Corresponding author. Tel.: +49-351-4646-2324; fax: +49-351-4646-2360/8711-612.E-mail address: gegenwart@cpfs.mpg.de (P. Gegenwart).                                                                                         and a strongly T -dependent specific heat coefficient C=T[3,4]. This led to the proposal that in these systemssuperconductivity is mediated by an electronic ratherthan the usual elastic Cooper pairing interaction [3].The cubic UBe13 is one of the most fascinating HFsuperconductors [5] because (i) superconductivitydevelops out of a highly unusual normal state charac-terized by a large and strongly T -dependent electricalresistivity [5,6] and (ii) upon substituting a small amountof Th for U in U1 xThxBe13, a non-monotonic evolu-tion of Tc and a second phase transition at Tc2 below Tc1,the superconducting one, is observed in a critical con-centration range xc1 ¼ 0:019 < x < 0:045 ¼ xc2 [7].In this paper, we investigate the low-temperaturenormal-state NFL behavior of a high-quality singlecrystal of UBe13 also studied in [8,9]. The T -dependencesobserved in specific heat (Section 2) and electricalresistivity (Section 3) resemble those found for Ce-basedHF systems near the antiferromagnetic (AF) QCP. Thisleads to the speculation that the NFL behavior in UBe13   158                                              is caused by a QCP at a finite magnetic field slightlyabove 4 T, covered by the superconducting (SC) phase.Indeed careful studies of the thermal expansion andspecific heat have revealed a ‘‘line of anomalies’’, BðT Þ,presumably of short-range AF origin, in the B–T phasediagram, that starts at TL ¼ 0:7 K (B ¼ 0) and termi-nates at Bð0Þ  4:5 T [9]. These anomalies have beenshown to mark the precursor of the lower lying of thetwo second-order phase transitions in the critical rangexc1 < x < xc2 in U1 xThxBe13 [9]. The B–T phase dia-gram presented in Section 4 suggests a relation betweenthe suppression of the TL anomaly in UBe13 within theSC state at 4.5 T and the normal-state NFL behavior.2. Specific heatThe specific heat has been measured with the aid of athermal-relaxation method at magnetic fields up to 12 T[10]. In Fig. 1, the electronic specific heat coefficient isdisplayed between 60 mK and 3 K. The high quality ofthe single crystal is reflected in the sharpness of thephase transition anomaly at Tc ¼ 0:9 K (10–90% width:25 mK). The equal areas construction [9] reveals a largereduced jump DCsc=cnTc of 2.6, when using cn ¼ 0:9 J/K2 mol as the electronic specific heat coefficient at Tc,that would indicate a strong-coupling SC state. How-ever, as can be seen from Fig. 1, in order to balance theentropy between the normal and SC states, the normal-state specific heat coefficient has to increase towardsT ! 0. A very similar observation has been made for therecently discovered Ce-based HF superconductor Ce-0.0 0.5 1.0 1.5 2.0 2.5 3.001234UBe 13C/T(J/K2mol)T (K)B(T)06810120.1 14012B(T)012Fig. 1. Specific heat C of a UBe13 single crystal after subtrac-tion from raw data the nuclear contribution due to Zeemansplitting of 9Be states [10], as C=T vs T at differing magneticfields. Solid line indicates the quasiparticle contributionassuming an axial SC order parameter [10]. Inset shows 0 and12 T data on a logarithmic temperature scale. Solid and dottedlines indicate C=T /  log T and C=T ¼ c0  bT 1=2, respec-tively.CoIn5 [11]. The application of magnetic fields whichsuppress superconductivity enables the study of thenormal state specific heat coefficient C=T down to lowertemperatures. Indeed an increase of C=T towards(1.4 ± 0.2) J/K2 mol is observed at B ¼ 12 T (inset Fig. 1.Here, the low-T upturn in the 12 T data below 0.2 K hasbeen ignored. The origin of the latter is unclear, since theexpected Schottky contribution due to the Zeemansplitting of the nuclear 9Be nuclear spin states has al-ready been subtracted from the raw data [10]. The originof the normal state logarithmic increase of C=T , regis-tered at B ¼ 12 T and resembling that observed forCeCoIn5 at B ¼ 5 T (Bc2) [11], will be discussed below.3. Electrical resistivityThe electrical resistivity of a small piece from thesame high-quality single crystal studied in specific heathas been measured by utilizing the standard four-ter-minal ac-technique [12]. As displayed in Fig. 2a, thezero-field resistivity qðT Þ displays a pronounced maxi-mum near Tmax  2 K. From the maximum value of theresistivity an inelastic mean free path as short as a few Acan be inferred, indicative of an incoherent metallicstate, that remarkably shows a rather low magneticsusceptibility. This had led Cox [6] to propose a quad-rupolar Kondo scenario for UBe13 by assuming a 4+(5f 2) Uranium valence state with a low-lying non-mag-netic C3 crystal-field (CF) doublet. This scenario revealsan intrinsic finite residual resistivity (at B ¼ 0) and alarge negative magnetoresistance slope dq=dB [13], asindeed observed, cf. Fig. 2a. On the other hand, CFeffects studied via specific heat [14] and Raman-scatter-ing [15] experiments as well as measurements of the non-linear susceptibility [16] seem to support a trivalent (5f 3)configuration. The resistivity maximum at 2 K indicates0 40501001502002501086415.51410864210B (T)UBe 13ρ (µΩcm)T (K)0.5 1.0 1.50.00.51.0baB (T)ρ/ ρ(1 K)1 2 3Fig. 2. (a) Electrical resistivity q vs T of a UBe13 single crystalat B ¼ 0 and differing fields; (b) the same data as in (a), nor-malized to the respective q (T ) value at T ¼ 1 K. Dashed line isan extrapolation to T ¼ 0 of the data for T P 0:8 K.                                             159a low-energy scale in addition to the characteristic‘‘Kondo-lattice’’ scale, T , accounting for the extremelylarge effective carrier masses. It has been estimated thatT  ranges from 8 K [14] to 30 K [17]. This additionalenergy scale manifests itself in a distinct maximum in thethermal expansion coefficient [18] and a less pronouncedshoulder in specific heat around 2 K and may be as-cribed to local spin fluctuations in disordered Kondosystems [9]. These ‘‘2 K fluctuations’’ dominate thenormal state resistivity for fields below 4 T. At largerfields, 4 T6B6 10 T, we are able to scale the variousqðT Þ curves within TcðBÞ6 T 6 1:2 K to a universalcurve by normalizing qðT Þ to its respective value at 1 K(Fig. 2b). Above T  0:7 K, the normal-state resistivityis found to be proportional to T , similar to what has beenobserved for optimally doped high-Tc cuprates [19]. Atlower temperatures, qðT Þ becomes superlinear.As demonstrated in Fig. 3a for the B ¼ 8 T data,DqðT Þ / T 3=2 for Tcð8 TÞ6 T 6 1 K. Apparently, this Tdependence is in full accord with the theoretical pre-diction for a three-dimensional system of itinerant AFspin fluctuations in the vicinity of a QCP [20,21]. Atelevated temperatures, the same scenario predicts across-over to a quasilinear T -dependence of qðT Þ, asindeed observed. As shown in the inset of Fig. 1, thespecific heat coefficient follows C=T /  log T forT P 0:3 K and gradually deviates to smaller values atlower T before the yet unexplained upturn sets in at thelowest temperatures. In the limited T range 0:15 K6T 6 0:4 K, the specific heat coefficient can be satisfac-torily described by C=T ¼ c0  bT 1=2 which is expectedalong with the T 3=2 behavior of DqðT Þ [20,21]. Thus, for4 T6B6 12 T both electrical resistivity and specificheat show NFL behavior consistent with the nearness ofan AF QCP, assuming three-dimensional critical AFfluctuations [20,21] and a substantial degree of potentialscattering [22].0.4 0.60.81 2T (K)1050100∆ρ(µΩcm)0.1 0.5 1 2T (K)151050100∆ρ(µΩcm)0 0.5 T 2 (K2)204060ρ (µΩcm)B = 15.5 TB = 8T T3/2 T3/2UBe13a bFig. 3. Temperature dependent contribution, Dq ¼ q  q0 ofthe electrical resistivity of a UBe13 single crystal vs T (on log-arithmic scales) at B ¼ 8 T (a) and B ¼ 15:5 T (b), respectively.Inset shows low-T data of (b) as q vs T 2.Since magnetic fields act on the critical spin-fluctua-tions associated with a magnetic QCP, a field-inducedchange from NFL to FL behavior is expected in thesesystems, as observed, e.g., at B0  6 T in ‘‘S-type’’CeCu2Si2 [23], B0  5 T in CeCoIn5 [24] and B0  1 T inCeNi2Ge2 [25]. For UBe13, this cross-over field is aslarge as 14 T at which DqðT Þ ¼ AðBÞT 2 becomes visiblebelow 0.3 K with a gigantic coefficient A. The latter isdecreasing with increasing B from 52 lX cm/K2 at 14 Tto 45 lX cm/K2 at 15.5 T [8,12], indicative for the de-crease of the quasiparticle–quasiparticle scattering cross-section upon tuning the system away from the QCP.Similar behavior has been observed in the other NFLsystems as well [23–25]. In YbRh2Si2, where weak AForder below TN ¼ 70 mK is suppressed at a criticalmagnetic field of Bc ¼ 0:06 T (applied in the easy mag-netic plane) a 1=ðB BcÞ divergence of AðBÞ has beenobserved very recently [26]. A more detailed analysis forUBe13 is, however, hampered by the SC phase and thelarge cross-over field B0  14 T. As for the other NFLsystems, in UBe13 the slope of the isothermal magneto-resistance changes from dqðBÞ=dB < 0 at B < B0 todqðBÞ=dB > 0 at B > B0 [12]. To summarize, UBe13shows striking similarities to other NFL systems withwell established AF quantum-critical points.4. B–T phase diagramAs shown in the previous sections, the normal-statebehavior of UBe13 is compatible with an AF QCP cov-ered by the SC phase. Measurements of the thermalexpansion coefficient aðT Þ on U1xThxBe13 have re-vealed, at B ¼ 0, a line of anomalies TLðxÞ in the T–xdiagram for x < xc1. TLðxÞ marks the precursor of thelower of the two phase transitions at Tc2, found for0:019 < x < 0:046 [9]. For pure UBe13 a negative peak inaðT Þ, clearly separated from the SC phase transition, hasbeen attributed to short-range AF correlations belowTL  0:7 K. An additional contribution that developsbelow TL is observed in the specific heat as well [10]. Thisbecomes visible upon comparing the measured C=T datawith the T dependence expected for the SC state (seesolid line in Fig. 1). The magnetic field dependence TLðBÞas determined from thermal expansion and specific heatis plotted in the B–T phase diagram, cf. Fig. 4. Mostimportantly, TL vanishes around 4.5 T, i.e. almost thesame field above which the universal NFL behavior inqðT Þ is observed. Thus, one may speculate that thepronounced NFL effects observed in the normal stateproperties are related to a field-induced QCP, TL ! 0 atabout 4.5 T. It is interesting to note, that a clearenhancement of the slope dTcðBÞ=dB is observed near 4T. Whether this stabilization of the SC state might berelated to this ‘‘short-range AF’’ QCP remains unclear.0 0.5 1 T (K)051015B (T)∆ρ  ~ T2dρ/dB > 0∆ρ  ~ T3/2dρ/dB < 0ρ ~ T"2K-fluctuations"Fig. 4. B–T phase diagram for UBe13. Open circles indicate SCphase boundary, defined by q ¼ 0, as derived from both T - andB-dependent measurements of the electrical resistivity [12]. So-lid symbols indicate ‘‘line of anomalies’’ BðT Þ obtained fromtemperature dependent thermal expansion (circles) and mag-netic field dependent specific heat (triangles) measurements,respectively [9]. Existence ranges for different power-lawbehaviors in qðT Þ as well as different signs of isothermal mag-netoresistance slope dqðBÞ=dB as discussed in text.160                                              5. ConclusionWe have discussed the low-T behavior of the HFmetal UBe13 for which an unconventional (yet not fullyidentified) SC ground state forms out an also uncon-ventional normal state. The latter can be studied only inrelatively large applied magnetic fields at lower tem-peratures, where it shows striking similarities withthe normal state of other NFL superconductors, e.g.‘‘S-type’’ CeCu2Si2 [23], CeCoIn5 [24] and CeNi2Ge2[25]. At B ¼ 0, thermal expansion studies have revealedan anomaly at TL  0:7 K, that appears, due to itsmagnetic field dependence and its negative sign, indica-tive of the freezing out of AF short-range correlations[9]. We propose that the low-lying 3D AF spin fluctua-tions responsible for the NFL properties in the normalstate of UBe13 are those associated with the field-in-duced suppression of the ‘‘TL-anomaly’’ at B  4:5 T.AcknowledgementWork at Florida was supported by the US Depart-ment of Energy, contract No. DE-FG05-86ER45268.References[1] J.G. Bednorz, K.A. M€uller, Z. Phys. B 64 (1986) 189.[2] F. Steglich et al., Phys. Rev. Lett. 43 (1979) 1892.[3] N.D. Mathur et al., Nature 394 (1998) 39.[4] H. Hegger et al., Phys. Rev. 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B 48 (1993) 7183.[21] T. Moriya, T. Takimoto, J. Phys. Soc. Jpn. 64 (1995) 960.[22] A. Rosch, Phys. Rev. Lett. 82 (1999) 4280.[23] P. Gegenwart et al., Phys. Rev. Lett. 81 (1998) 1501.[24] J. Paglione et al., Phys. Rev. Lett. 91 (2003) 246405.[25] P. Gegenwart et al., Phys. Rev. Lett. 82 (1999) 1293.[26] P. Gegenwart et al., Phys. Rev. Lett. 89 (2002) 056402.

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Non-Fermi liquid normal state of the Heavy Fermion superconductor UBe13

Non-Fermi liquid normal state of the Heavy Fermion superconductor UBe13

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