Muon spin rotation and non-Fermi liquid behavior in UCu4Pd

Muon spin rotation and non-Fermi liquid behavior in UCu4Pd

Physb=5803=Jayashree=Venkatachala=BG Physica B 289}290 (2000) 15}19 Muon spin rotation and non-Fermi liquid behavior in UCu Pd  D.E. MacLaughlin ...

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Physica B 289}290 (2000) 15}19

Muon spin rotation and non-Fermi liquid behavior in UCu Pd 

D.E. MacLaughlin *, R.H. He!ner, J.E. Sonier, G.J. Nieuwenhuys, R. Chau, M.B. Maple, B. Andraka, G.M. Luke, Y. Fudamoto, Y.J. Uemura, A. Amato, C. Baines Department of Physics, University of California, Riverside, CA 92521-0413, USA MS K764, Los Alamos National Laboratory, Los Alamos, NM 87545, USA Kamerlingh Onnes Laboratory, Leiden University, 2300 RA Leiden, The Netherlands Department of Physics, University of California, San Diego, La Jolla, CA 92093, USA Department of Physics, University of Florida, Gainesville, FL 32611, USA Department of Physics and Astronomy, McMaster University, Hamilton, Ont., L8P 4N1, Canada Department of Physics, Columbia University, New York, NY 10027, USA Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland

Abstract Muon spin rotation and relaxation (lSR) experiments have been carrried out on the non-Fermi liquid (NFL) heavy-fermion compound UCu Pd. Transverse-"eld lSR line widths indicate that the U-ion susceptibility is in homogeneous and in good agreement with predictions of disorder-driven NFL mechanisms. The inhomogeneity seems to be remarkably independent of the exact amount of structural disorder. Longitudinal-"eld experiments yield muon spin-lattice relaxation rates which are two orders of magnitude faster than estimates from a single-ion &Kondo disorder' model. This suggests a cooperative rather than a single-ion NFL mechanism. No evidence is found for freezing of more than a few percent of U-ion moments in UCu Pd above 100 mK.  2000 Elsevier Science B.V. All rights reserved.  Keywords: Non-Fermi liquid; UCu Pd ; Kondo disorder; Gri$ths phase \V V

1. Introduction Heavy-fermion alloys in the series UCu Pd , \V V 1:x:1.5, exhibit low-temperature thermodynamic and transport properties which are not in agreement with the expected Landau Fermi-liquid predictions [1,2]. Many such non-Fermi liquid * Corresponding author. Tel.: #1-909-787-5344; fax: #1909-787-4529. E-mail address: [email protected] (D.E. Mac Laughlin).  Present address: Lawrence Livermore National Laboratory, Livermore, California 94550, USA.

(NFL) heavy-fermion materials are close to a magnetic region of the appropriate phase diagram, and in addition are disordered due to point defects (chemical substitution, site interchange, etc.). Theories of NFL behavior often relate one or both of these features to the driving mechanism. Nuclear magnetic resonance (NMR) and muon spin rotation (lSR) experiments in UCu Pd \V V alloys [3}6] revealed static line widths which vary much more rapidly with temperature than the bulk susceptibility and which become anomalously large at low temperatures. This result motivated Bernal et al. [3] to analyze their NMR data in terms of

0921-4526/00/$ - see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 0 ) 0 0 2 3 3 - 7

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a &Kondo disorder' model [7,8], in which disorder in the hybridization between conduction electrons and localized f-electron moments results in a wide distribution of Kondo temperatures ¹ . At a given ) temperature ¹ local moments with ¹ (¹ are ) uncompensated and cause the NFL behavior. The wide spread of Kondo temperatures gives rise to a temperature-dependent spread ds of local-moment susceptibilities s(¹, ¹ ) which is re#ected in ) the NMR and lSR line widths. Recently Castro Neto et al. [9] described a model in which the combination of disorder and a critical point associated with cooperative behavior of interacting local moments gives rise to a &Grif"ths phase' of correlated clusters, with critical behavior described by a nonuniversal scaling exponent j. This picture agrees well with thermodynamic data for a number of NFL systems [10]. The present paper presents evidence from lSR experiments concerning two aspects of NFL behavior in UCu Pd. We conclude that  E in spite of a possible ordered structure for stoichiometric UCu Pd there is su$cient dis order for a disorder-driven mechanism to be applicable, and E the behavior of muon spin-lattice relaxation in UCu Pd rules out the single-ion (Kondo dis order) scenario in favor of a cooperative mechanism, although a quantitative prediction of the Gri$ths-phase theory [9] is not born out by the experimental results.

2. Disorder-driven NFL behavior in UCu4 Pd UCu Pd , 0)x:2.5, crystallizes in the FCC \V V AuBe structure (space group F4 3 m). The end  compound UCu possesses two crystallographi cally inequivalent copper sites in the ratio 4 : 1 at the 16e and 4c positions (Wycko! notation). Substitution of Pd for Cu could cause stoichiometric UCu Pd to crystallize as shown in Fig. 1, in which  case disorder might not play a role in the NFL behavior of this compound. Recently Chau et al. [10] reported elastic neutron di!raction measurements on members of the UCu Pd series. From Rietveld re"nements of \V V

Fig. 1. Unit cell of UCu Pd assuming structural order, with  atom sites indicated in Wycko! notation. Muon stopping sites in the end compound UCu (Refs. [11,12]) are also shown. 

the data they "nd that for x"1 &2there is no evidence for Pd/Cu disorder2' [13]. But the diffraction data do not rule out the possibility of site interchange between Pd and Cu sites at the level of &4% occupation of 16e (Cu) sites by Pd atoms (and therefore &16% occupation of 4c (Pd) sites by Cu atoms). We have carried out lSR studies of a number of samples of UCu Pd, including a previously-studied  [5,6] powder sample (Powder C1), the powder sample used for the neutron di!raction experiments (Powder C2), and a bulk polycrystal consisting of a few single crystals. Some of these results have been reported previously [4}6,14]. TF-lSR data were obtained at the M20 beam line at TRIUMF, Vancouver, Canada. The mean 1K2 and width dK of the positive-muon (l>) frequency shift distribution were obtained from "ts of the relaxation function G(t)"A exp(!c H dKt) l  ;cos[c H (1#1K2)t# ], l 

(1)

to time-di!erential l> relaxation data. Here A is the initial l> decay asymmetry, c is the l> gyrol magnetic ratio, H is the applied transverse "eld,  and is the phase of the initial l> spin orientation.

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Fig. 2. Dependence of dK/(aHs) on bulk susceptibility s, with temperature an implicit parameter, in UCu Pd. Powder C1:  previously studied sample (Refs. [5,6]). Powder C2: present sample, used in neutron di!raction studies (Ref. [13]). Curve: ds/s from the Kondo disorder theory (Refs. [4}6]).

Estimates of the relative spread ds/s were obtained using the relation ds dK " , aHs s

(2)

where aH is an e!ective l>/U-ion dipolar coupling constant which can be calculated from lattice sums [4}6]. Fig. 2 plots dK/(aHs) versus the uniform bulk susceptibility s, with temperature (2}300 K) an implicit parameter. The reproducibility of dK/(aHs) between all three samples is striking. The three samples were all prepared di!erently and are likely to have a range of defect concentrations, in which case the susceptibility inhomogeneity is remarkably insensitive to the amount of disorder. Also shown in Fig. 2 for comparison is ds/s from the Kondo disorder model [4]. The disorder-driven Gri$thsphase theory (not shown), is also in good agreement with the data [9].

3. Muon relaxation and NFL behavior in UCu4 Pd Zero- and longitudinal-"eld lSR data were obtained at the LTF facility, Paul Scherrer Institute (PSI), Villigen, Switzerland, using a powder sample of UCu Pd (Powder C2). A preliminary "t was  made to a &stretched exponential' relaxation func-

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Fig. 3. Temperature dependence of characteristic relaxation rate K ("lled circles) and power b (open circles) from "ts of zeroand longitudinal-"eld lSR data to the stretched-exponential form G(t)"exp[!(Kt)@] in UCu Pd. (a) Longitudinal "eld  H "0. (b) H "300 Oe. * *

tion G(t)"exp[!(Kt)@], which parameterizes an inhomogeneous distribution of (exponential) relaxation rates if the power b is less than 1. The temperature dependence of the relaxation rate K and b is given in Fig. 3 for longitudinal "eld H "0 and 300 * Oe. The fact that H "300 Oe is unable to sup* press (i.e., &decouple') the l> relaxation indicates that at least for this "eld G(t) re#ects spin-lattice relaxation due to thermal #uctuations of U-ion moments. The broad distribution indicated by the low value of b (&0.4}0.7 below 1 K) is qualitatively consistent with the spatial inhomogeneity inherent to both the Kondo disorder and the Gri$ths-phase pictures. Even though the stretched-exponential "t is by no means unique, several issues are addressed by these results. The data can be compared with the single-ion Kondo disorder model, which predicts inhomogeneous relaxation due to the distributed ¹ . For a distribution function P(¹ ) this mecha) ) nism yields a spatially averaged relaxation rate 1K2"d¹ P(¹ )K(¹ ) roughly given by ) ) )

cB 1K2+ l , k 1¹ 2 )

(3)

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where B is the mean-square #uctuating dipolar "eld at the l> site and 1¹ 2 is the spatially aver) aged Kondo temperature. With 1¹ 2+50}100 K ) for UCu Pd [3,4] we estimate 1K2+1!  2;10\ ls\, which is two orders of magnitude smaller than the observed values (Fig. 3). We conclude that the single-ion Kondo disorder mechanism is not capable of explaining the relaxation data. The results are also relevant to the question of static ordered or spin-glass magnetization [15,16]. Above &100 mK our data show no sign of the large increase of K which would be expected from freezing of more than a few per cent of U-ion moments. The data suggest slowing down of the U spins as ¹P0, rather than a transition at a nonzero temperature. Finally, we consider whether the data are consistent with a cooperative relaxation mechanism. In the Gri$ths-phase model [9] the imaginary component s(u, ¹) of the dynamic spin susceptibility is given by s(u, ¹)Ju\>H tanh( u/k ¹), consistent with the inelastic neutron scattering results of Aronson et al. [17] in UCu Pd with j+. The \V V  standard relation between s(u, ¹) and the l> relaxation rate then gives ¹ KJ s(u , ¹)Ju\>H, l l u l

(4)

since u ;k ¹. This is independent of temperl ature, contrary to the data. It is conceivable, however, that the simplest Gri$ths-phase model does not treat s(u, ¹) properly for the very low l> frequency u ;k ¹/ . Eq. 4 predicts a "eld dependl ence KJH\>H since u JH . Preliminary data * l * (not shown) are consistent with this "eld dependence, also with j+, but are not conclusive. 

4. Conclusions Our lSR studies indicate that a cooperative disorder-driven mechanism is the most likely source of NFL behavior in UCu Pd, and that no evidence is  found for freezing of more than a few percent of

U-ion moments above 100 mK. Future lSR experiments will be necessary to characterize the lowfrequency spin #uctuations in UCu Pd more  completely, and to determine whether disorderdriven NFL behavior plays a role in other nominally ordered compounds.

Acknowledgements This research was supported in part by the US National Science Foundation, Grants DMR9731361 (U.C. Riverside), DMR-9705454 (U.C. San Diego), DMR-9400755 (U. Florida), and DMR9510454 (Columbia), and by the U.C. Riverside Academic Senate Committee on Research, the Netherlands agencies NWO and FOM (Leiden), and the Japanese agency NEDO (Columbia), and was carried out in part under the auspices of the US DOE (Los Alamos).

References [1] B. Andraka, G.R. Stewart, Phys. Rev. B 47 (1993) 3208. [2] R. Chau, M.B. Maple, J. Phys.: Condens. Matter 8 (1996) 9939. [3] O.O. Bernal, D.E. MacLaughlin, H.G. Lukefahr, B. Andraka, Phys. Rev. Lett. 75 (1995) 2023. [4] D.E. MacLaughlin, O.O. Bernal, H.G. Lukefahr, J. Phys.: Condens. Matter 8 (1996) 9855. [5] O.O. Bernal et al., Phys. Rev. B 54 (1996) 13 000. [6] D.E. MacLaughlin et al., Physica B 230}232 (1997) 606. [7] E. Miranda, V. DobrosavljevicH , G. Kotliar, J. Phys: Condens. Matter 8 (1996) 9871. [8] E. Miranda, V. DobrosavljevicH , G. Kotliar, Phys. Rev. Lett. 78 (1997) 290. [9] A.H. Castro Neto, G. Castilla, B.A. Jones, Phys. Rev. Lett. 81 (1998) 3531. [10] M.C. de Andrade et al., Phys. Rev. Lett. 81 (1998) 5620. [11] S.R. Barth et al., Hyper"ne Interact. 31 (1986) 397. [12] S.R. Barth et al., J. Magn. Magn. Mater. 76&77 (1988) 455. [13] R. Chau, M.B. Maple, R.A. Robinson, Phys. Rev. B 58 (1998) 139. [14] D.E. MacLaughlin et al., Phys. Rev. B 58 (1998) R11 849. [15] R. Volmer et al., Physica B 230}232 (1997) 603. [16] E.-W. Scheidt et al., Phys. Rev. B 58 (1998) R10 104. [17] M.C. Aronson et al., J. Phys.: Condens. Matter 8 (1996) 9815.

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Comments Kalvius: Another scenario for NFL behaviour is the multi-channel Kondo ewect. Can this be excluded in your case?

MacLaughlin: At least in its simplest form, the multichannel Kondo ewect requires special crystal symmetry (e.g. cubic symmetry) to produce the required nonmagnetic ground state. It is hard to see how this condition would survive disorder.

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He4ner: Do you have a qualitative picture for the yuctuation in ;Cu Pd  becoming more homogeneous (b P 1) as you cool below 1 K?

MacLaughlin: Only the speculation that the lowest-lying excitations of the system might be extended and thus they average the disorder somewhat. But there is no independent evidence for this.