Heavy-fermion formation in USn3: Static and dynamical properties
S. Kambe,1 H. Sakai,1 Y. Tokunaga,1 T. D. Matsuda,1 Y. Haga,1 H. Chudo,1 and R. E. Walstedt21Advanced Science Research Center, Japan Atomic Energy Agency, Tokai-mura, Ibaraki 319-1195, Japan
2Physics Department, University of Michigan, Ann Arbor, Michigan 48109, USAReceived 13 December 2007; revised manuscript received 14 February 2008; published 7 April 2008
USn3 is a heavy-fermion system with an electronic specific heat coefficient 170 mJ /K2 mol. In order tofurther characterize the heavy-fermion phenomena for USn3, the Knight shift and spin-lattice relaxation timeT1 of
119Sn NMR have been measured. The static specific heat and dynamical T1 properties in the heavy-fermion state can be described in a quantitatively consistent way in terms of a spin-fluctuation model with twoconstant energy scales. However, it is necessary to introduce a T-dependent effective RudermanKittelKasuyaYosida interaction JQ JQa+bT in order to describe the crossover from an incoherent, localizedstate to a coherent, heavy-fermion state. In addition, a universal scaling behavior is proposed for the crossoverregime. The parameters obtained are used to predict the T dependence of the thermal expansion coefficient.
DOI: 10.1103/PhysRevB.77.134418 PACS numbers: 76.60.k, 75.30.Mb
In f-electron itinerant systems, heavy-fermion HF stateshave been observed at low temperatures in certaincompounds.1 In such HF systems, the f-electron momentsare localized above the effective Fermi temperature T*. AtTT*, a crossover from the incoherent localized state to thecoherent HF state takes place, driven by the Kondo interac-tion between the localized f moments and the conductionelectrons, which usually predominates over the RudermanKittelKasuyaYosida RKKY interaction between thef-electron moments. In the HF state, the static magnetic sus-ceptibility 0,0 and the Sommerfeld electronic specificheat coefficient =Cel /T become quite large compared withordinary metals.
The T dependence of the dynamical susceptibilityIm q , also clearly reflects the formation of a HF state.For example, in CeRu2Si2,
2 the spin-lattice relaxation rate1 /T1T Im q , increases with decreasing T and becomesconstant below TT*. In ordinary metals, 1 /T1T constantbehavior is observed for a wide temperature range Korringabehavior. In contrast, in HF systems, such behavior is ob-served when an HF state i.e., Fermi liquid is formed belowT*. At temperatures well above T*, T1 becomes constant,which is characteristic behavior for systems with localizedmoments.
Up to now, the static uniform susceptibility, specific heat,etc. and dynamical spin-lattice relaxation time, inelasticneutron scattering, etc. properties have been discussed sepa-rately in HF systems. In this study, we have tried to repro-duce the T dependence of the specific heat and spin-latticerelaxation time of the HF system USn3 in a quantitativelyconsistent way using the spin-fluctuation model for itinerantmagnets by Moriya.3 This approach gives a good account ofboth quantities for temperatures below T*. However, T de-pendence of the effective RKKY interaction has to be in-troduced in order to describe the crossover regime.
In this paper, data for the T dependences of the Knightshift and spin-lattice relaxation time at the Sn site in USn3are presented. Among UX3 compounds with the same AuCu3fcc structure, where X is a IVB element X :Si ,Ge,Sn,Pb,USn3 shows the largest
4,5 170 mJ /K2 mol and 0,0=9.0103 emu /mol at T0 K. These parameters give a
Wilson ratio of 2, indicating that USn3 is a typical heavy-fermion compound.
A powder USn3 sample was prepared for NMR measure-ments by crushing a well-characterized single crystal.6 Theresistivity of the sample showed good metallic behavior witha residual resistivity of 1.7 cm. 119Sn I=1 /2 NMRmeasurements were performed using a conventional pulsedspectrometer with a 12 T superconducting magnet. The /2 pulse sequence has been used to excite nuclear spin-echo signals. Field-sweep NMR spectra were taken at a fre-quency of 119 MHz using digital averaging of the nuclearspin-echo signals. Spin-lattice relaxation time T1 data werealso obtained with this method.
III. EXPERIMENTAL RESULTS
A. Static susceptibility
Figure 1 shows the cubic AuCu3-type crystal structureof USn3. The U site has cubic local symmetry, whereas forthe Sn site, it is tetragonal. The local principal axis for the Sn
FIG. 1. Color online Crystal fcc structure of USn3. n indi-cates the local symmetry axis of the Sn site.
PHYSICAL REVIEW B 77, 134418 2008
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site indicated as n in Fig. 1 is important for analyzing theNMR results.
Figure 2 shows the T dependence of the static susceptibil-ity 0,0 of the NMR sample. At high temperatures, 0,0shows CurieWeiss CW behavior see inset. Below 30 K,0,0 starts to saturate and finally becomes independent ofT below 6 K. The latter behavior signals the formation of aHF state. Thus, we consider that T*30 K in USn3. Thedata for 0,0 have been least-squares fitted to the CWfunction 0,0=const0,0+
3kBTabove 50 K, yielding
const0,0=3.0105 emu /mol, an effective moment ef f=2.44B, and a Weiss temperature =58 K. The constantterm const0,0 usually represents the Van Vleck orbital anddiamagnetic susceptibilities. The value obtained forconst0,0 is quite small compared with 0,0, indicatingthat the dominant term in USn3 is the spin susceptibilityspin0,0. In the inset to Fig. 2, the T dependence of1 / 0,0const0,0 is presented. The straight line ob-tained above 50 K confirms an ideal CW behavior at hightemperatures. The effective moment eff=2.44B is rathersmaller than the U3+ ionic value 3.87B, indicating that the5f electrons have an itinerant nature below 300 K.
In band calculations, the noninteractive Pauli spin suscep-tibility is estimated as 6.6104 emu /mol from the densityof states at the Fermi level,7 yielding a 14 enhancement of0,0 in the HF state.
B. NMR spectrum and Knight shift
Figure 3 shows a field sweep spectrum at an NMR fre-quency of 119 MHz for the 119Sn at 1.6 K. This is a typicalpowder-pattern spectrum for I=1 /2 in a site of tetragonaluniaxial symmetry. The Knight shifts K for applied mag-netic field H n and K for H to the principal axis n of theSn site are determined based on fitting procedures usingan axially symmetric powder pattern with Gaussianbroadening.8 From K and K, the isotropic and anisotropiccomponents, KisoK +2K /3 and KaniK K /3,have been estimated.
Figure 4 shows the T dependence of Kiso and Kani. SinceKisoKani, the overall shift is basically isotropic in this com-pound. Kiso and Kani are plotted vs 0,0 the so-called K-plots in Fig. 5, showing good linearity in both cases. Hyper-fine coupling constants A0iso=68 kOe /B for Kiso andA0ani=8.8 kOe /B for Kani are determined from the slopesof linear fits to these data Table I. Since hyperfine cou-plings of this magnitude cannot be explained by classicaldipolar-dipolar interactions, transferred hyperfine fields dueto hybridization between U 5f and Sn 5s, p orbitals are con-sidered to be the underlying mechanism. Since Kiso is mainlydriven by spin polarization transferred to the Sn 5s orbital,the point on the plot where Kiso=0 corresponds to where thestatic spin susceptibility spin0,0=0. As shown in Fig. 5,the extrapolation of the Kiso-0,0 plot very nearly inter-
FIG. 2. Color online T dependence of the static susceptibility0,0. In the inset, a CurieWeiss plot of 1 / 0,0const0,0vs T is presented.
FIG. 3. Color online 119Sn field sweep NMR spectrum at119 MHz in a powder sample of USn3 T=1.6 K. The edge posi-tions indicated by arrows correspond to shift parameters K H nand K Hn. Solid line is a fitted curve based on the axiallysymmetric powder pattern with Gaussian broadening.
FIG. 4. Color online T dependence of the isotropic Kisoand anisotropic Kani Knight shifts KisoK +2K /3 and KaniK K /3.
FIG. 5. Color online Knight shift versus static susceptibility0,0 plot K- plot. The solid lines were obtained with least-squares fits. The slopes of the lines correspond to the hyperfinecoupling constants A0iso and A0ani.
KAMBE et al. PHYSICAL REVIEW B 77, 134418 2008
sects the origin, indicating that 0,0spin0,0 in USn3.This is consistent with the small nonspin susceptibility constwhich was estimated from the static susceptibility measure-ments. The origin of Kani is apparently hybridization with theSn 5p orbital, which gives an anisotropic, dipolar hyperfinefield. The large Kiso compared with Kani indicates that thehybridization between the U 5f and Sn 5s orbitals is compa-rable with that between the U 5f and Sn 5p orbitals since the5s hyperfine field is expected to be much larger than that ofthe 5p electrons.
Since the K- plot is linear down to 1.6 K in the HF state,the hyperfine coupling constant at the Sn site is not modifiedby the HF effects in this compound. In some HF compoundsthe K- plot becomes nonlinear due to T-varying hyperfineinteraction contributions.9 In contrast, in USn3 there is nosuch complication from T-dependent couplings; thus, a quan-titative analysis is possible in detail as described below.
C. Spin-lattice relaxation time T1
Figure 6 shows the T dependence of 1 /T1 at the 119Snsite for Hn measured at the K position in Fig. 3.
At high temperatures, the T dependence of T1 becomesweak. This regime corresponds to a crossover from the HF toa localized momentlike state. In the localized moment state,T1 is independent of T and is expressed as the exchange-narrowing limit case,10
1/T1ex = 2A0/2JJ + 13nex
Here, the effective total angular momentum J and ex-change frequency ex are estimated to be 0.82 and3.11012 s1, respectively, from the CW parametersef f =2.44B and =58 K, and n is the number of neigh-boring magnetic sites around Sn, i.e., n=4 in the presentcase. These parameters lead to 1 /T1ex1.9105 s1, whichis much larger than the observed value at 240 K. Even at
high temperatures, the 5f electrons still have some itinerantcharacter in USn3.
At low temperatures, 1 /T1 is proportional to T below10 K, i.e., Korringa behavior appears in the HF state. Foritinerant systems with magnetic exchange enhancements,the usual Korringa relation between T1, K and Se /N2h /4kB: T1TK2=S is modified with K term,leading to an extended Korringa relation for ligand sites,11
T1TK2 = nSK1. 2
The estimated K0.6 is smaller than 1 in USn3, suggest-ing that ferromagnetic exchange enhancement seems to bepresent in this compound. However, an alternative explana-tion is the cancellation of antiferromagnetic fluctuations atthe 119Sn, as discussed below. The present case indicates thatwe should be careful in applying the extended Korringa re-lation to ligand-site NMR results when the hyperfine formfactor is a critical element. In fact, in the paramagnetic stateof the isostructural antiferromagnet UIn3 TN=88 K, K isfound to be less than 1 owing to a similar cancellation.12
In order to estimate the anisotropy of hyperfine fluctua-tions, 1 /T1 for H n has also been measured at 20 and120 K not shown. Generally, 1 /T1, at tetragonal sites isexpressed as
Aq2 Im q,n
fq2A02 Im q,n
qAq2 Im q,nn
+ Aq2Im q,n
2 q fq2A02 Im q,n
+ fq2A02 Im q,n
where N is the nuclear gyromagnetic ratio, Aq is the hy-perfine coupling constant, fq is the hyperfine form factorwhich reflects the local symmetry of Sn site, Im q ,n isthe dynamical susceptibility, and n=119 MHz is the NMRfrequency. The off-diagonal hyperfine coupling13 is ignoredhere since the hyperfine coupling is very nearly isotropic atthe Sn site. Thus, the hyperfine form factor fq is the samefor H n and Hn .
Based on Eq. 3,
TABLE I. Transferred hyperfine coupling constants A0 inkOe /B for the 119Sn in USn3 obtained from K- plots.
A0 A0 A0iso A0ani
862 592 682 8.80.5
FIG. 6. Color online T dependence of 1 /T1 at the 119Sn site forHn .
HEAVY-FERMION FORMATION IN USn3: STATIC PHYSICAL REVIEW B 77, 134418 2008
fq2 Im q,n
fq2 Im q,n. 4
The experimental value of21/T11/T1
1/T120.2 at 20 and
120 K then arises from the anisotropy of the transferred hy-perfine coupling constant A02 /A0
2 2.1 see Table I.From Eq. 4, this fact indicates that the dynamicalsusceptibility Im q , is isotropic i.e., Imq ,n=Im q ,n, in agreement with the local cubic symmetryof the U site, which makes the main contribution to the mag-netic susceptibilities i.e., 0,0 and Im q ,.
IV. ANALYSIS AND DISCUSSION BASED ON THE SCRMODEL OF DYNAMIC MAGNETISM
A. Analysis based on the usual self-consistentrenormalization model
In this section, the electronic specific heat4 and the presentspin-lattice relaxation results are interpreted in a quantita-tively consistent way based on the framework of the self-consistent renormalization SCR model. Originally, the SCRmodel was developed by Moriya in order to interpret weakmagnetism in itinerant systems.3 More recently, this modelwas adapted to describe the HF state.14
In the SCR model, the dynamical susceptibility is charac-terized by two energy scales, T0 and TA, which correspond tomagnetic fluctuation energy in and q spaces, respectively.The q dependence of the RKKY interaction JQ is expressedas JQJQ+q=2TAq / qB 2 around the antiferromagneticwave vector Q, where qB is the zone-boundary vector. Thus,the dispersion of the RKKY interaction can be defined asJQJQJ0=2TA. Since this JQ includes all q-dependenteffects such as s f mixing in addition to the original RKKYinteraction,14 JQ represents effective RKKY interaction. Usu-ally, T0 has a magnitude of T* and is connected with the localmagnetic susceptibility. The framework of the SCR modelfor application to HF systems has been described in previousworks.15,16
Since no magnetic phase transition takes place in USn3,the nature of the magnetic correlations must be decided insome other way. In the present analysis, correlations havebeen presumed to be antiferromagnetic see below.
For the case of antiferromagnetic correlations, the generalsusceptibility Q+q , has been formulated to be14
Q + q,=
= 2TAyT + qqB2 i2T0 ,yT
Q q 0 ,
JQ =L Q
where B is the dispersion constant, L and q are the localand q-dependent susceptibilities, and L and q are the localand q-dependent relaxation rates, respectively. The T depen-dence of yT is determined based on the fluctuation-dissipation theorem in order to guarantee internal consis-tency on condition that the total spin fluctuation S2 isconstant.3
In addition, two dimensionless parameters y0 and y1 areintroduced to characterize the state of the system,
y0 yT = 0 K, y1 2JQ
Here, y0 is a measure of the deviation from the quantumcritic...