Low Frequency Spin Dynamics in the CeMIn Materials 5 N. J. Curro, J. L. Sarrao, and J. D. Thompson Condensed Matter and Thermal Physics, Los Alamos National Laboratory, Los Alamos, NM 87545, USA P. G. Pagliuso Instituto de F´ısica “Gleb Wataghin”, UNICAMP, 13083-970, Campinas-SP, Brazil 3 Sˇ. Kos, Ar. Abanov, and D. Pines 0 Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA 0 (Dated: February 1, 2008) 2 WemeasurethespinlatticerelaxationoftheIn(1)nucleiintheCeMIn materials,extractquanti- n 5 a tativeinformationaboutthelowenergyspindynamicsofthelatticeofCemomentsinbothCeRhIn5 ∗ J and CeCoIn5, and identify a crossover in thenormal state. Abovea temperature T theCe lattice exhibits ”Kondo gas” behavior characterized by local fluctuations of independently screened mo- 8 ments; below T∗ both systems exhibit a ”Kondo liquid” regime in which interactions between the ] local moments contribute to the spin dynamics. Both the antiferromagnetic and superconducting l groundstatesinthesesystemsemergefromthe”Kondoliquid”regime. Ouranalysisprovidesstrong e - evidencefor quantumcriticality in CeCoIn5. r t s PACSnumbers: 76.60.-k,75.20.-g,75.25.+z,71.27.+a . t a m The CeMIn5 heavy fermion materials possess a rich cal fluctuations of the Ce moments. Below T∗ the q- - phase diagram revealing a fascinating interplay between independent local moment contribution to T1−1 becomes d the antiferromagnetic behavior of incompletely screened temperatureindependent, andandasecondqandT de- n Ce local moments, an unconventional normal state, and pendent component emerges. We identify this emergent o superconducting behavior of the heavy electrons that is component with the heavy electrons, that is the itiner- c [ reminiscentofthatfoundinthecuprates[1,2]. Thenon- ant component of the Ce 4f electrons arising from their Fermiliquidbehavioroftheitinerantheavyelectronsand couplingto one anotherandto the conductionelectrons. 2 the possible d-wave symmetry of their superconducting This extra contribution to the relaxation agrees quanti- v state has led to the proposal that the superconducting tatively with inelastsic neutron scattering (INS) results 4 5 pairingmechanismarisesfromtheircouplingtospinfluc- inCeRhIn5,andsuggeststhattheantiferromagneticcor- 3 tuations [1, 3, 4, 5, 6]. In this Letter, we consider the relations in CeCoIn5 are primarily 2D, and that in the 5 information about the coupled system of Ce local mo- absence of superconductivity one would have a quantum 0 ments and itinerant heavy electrons that can be derived critical point corresponding to a transition from Fermi 2 from Nuclear Magnetic Resonance (NMR) experiments liquid to antiferromagnetic behavior at T =0. 0 on the spin lattice relaxationrate (T−1) of the In(1) nu- / 1 In orderto developa model for the hyperfine coupling t clei located in the plane of the Ce moments. We show a intheCeMIn5 materials,itisnecessarytodeterminethe that the In(1) nuclei are strongly coupled to their four m number of spin degrees of freedom, and the hyperfine nearest neighbor Ce spins by an anisotropic hyperfine - couplings to each degree of freedom. For all three mate- d interaction, and that the resulting anomalous behavior rials (M=Rh, Ir, Co), the bulk susceptibility χ is dom- n of T1T provides important information on the low fre- inated at high temperatures by localized Ce moments, o quencydynamicsofthelatticeofCespins,andthequasi- and can be adequately fit by an expression for these c particles to which they couple. Because the coupling is : moments in a tetragonal crystalline electric field (CEF) v anisotropic, it does not vanish for antiferromagnetically i correlated Ce moments, so that T−1 provides valuable [11, 12, 13, 14, 15]. Neutron diffraction (ND) in the or- X 1 dered state of CeRhIn5 indicates that the magnetism is information on the dynamics of their magnetic ordering, r localized on the Ce sites, albeit with reduced moments as well as on the influence of Kondo screening of the a [16]. Thereducedmomentplausiblyreflectssomedegree moments on their relaxation rate, and departures from ofhybridizationwithincompletescreeningoftheCemo- Kondo behavior at low temperatures [7, 8]. We present ment in the CEF ground state doublet. (For the CEF our results for both the antiferromagnet CeRhIn5 and scheme measured in [15] the ground state doublet has the superconductor CeCoIn5. a moment of 0.92µ , whereas the measured moment is B Previous reports of T1−1 in CeIrIn5 and CeCoIn5 by 0.74µB [16, 17, 18].) The magnetic shifts of the high other authors have attributed an unusual temperature symmetry In(1) site in CeRhIn5, presented in Fig. (1), dependence of T−1 in the normal state to antiferromag- are linear in χ in the paramagnetic state. If the main 1 netic fluctuations up to room temperature [9, 10]. We contribution to the magnetic shift arises from the local- show in both CeRhIn5 and CeCoIn5 that above a tem- ized Ce moments, then Kα = K0,α+βαχα, where βα is perature T∗ of the order 10K, T−1 is dominated by lo- an effective hyperfine coupling and α = c,ab. Note that 1 2 0.16 1000 8 1) 7 (a) -K 6 (b) 4ξ/a H || c 20 0.14 -1c 4 2 6 6 0 0.12 -1T) (se 1002 410 K (%)c2 K (%)ab3 K (%) 864 1150 (x10 emu/mχ-3 TT ( sec K)1 000...100086 (T11.50.46001 T-T0N.1 (K) 1021 o 2 2 l) V)1.0 0 1 H0 || ab 00 χ5 (x1100-3 e1m5u/2m0ol2)5 5 0.04 Γ (me0.5 0.02 -2 0 (c) 0 0 10 20 30 40 50 0.00 0.0 0 20 40 60 80 0 20 40 60 80 T (K) T (K) T (K) FIG. 1: Themagnetic shift of In(1) in CeRhIn (circles) and 5 FIG. 2: (a) T T versus T in CeRhIn and CeCoIn ; the the bulk susceptibility (lines) versus temperature. The in- 1 5 5 CeRhIn data for T > 20K are taken from [20] (CeRhIn set shows Kα versus χα. The solid (open) circles and solid NQR (•5), CeCoIn H ||c (solid squares), and H ||ab (open5 (dashed) lines are for H||ab (H||c). squares)). (b) (T T5)−10−111.43sec−1K−1 (•) and0ξ/a (open 1 squares, ref [7]) vs. T −TN in CeRhIn5, where the solid line isafitasdescribedinthetext. (c)Γvs. T forCeRhIn . The 5 the hyperfine coupling is anisotropic (β = 26.4kOe/µ , c B dashedlineisalinearfittothedataabove20K,andthesolid βab =19.6kOe/µB), and the size of these hyperfine cou- line is taken from [8] using T0 =5K. plings aremorethananorderofmagnitude greaterthan typicaldipolarfields(∼1kOe/µ ),sotheIn(1)mustbe B coupledtotheCeviaananisotropictransferredhyperfine field Hint at the In(1) site for T << TN is 1.7 kOe [19], coupling of electronic origin. which corresponds to a value x=0.12. We postulate thatthe In(1) nucleus is coupledto each The spin lattice relaxationof the In(1) nuclei is deter- of the four nearest neighbor Ce spins through a tensor mined by the fluctuations of the Ce spins, and is given A˜ = (cid:8)A||,A⊥,Ac(cid:9), where the principal axes lie parallel by the Fourier component of hH(t)H(0)i at the Larmor and perpendicular to the Ce-In(1) bond axis in the ab frequency, where the time dependence arises from the plane, and along the c axis: fluctuations of the Ce spins S . Moriya showed that the i spin lattice relaxation rate can be expressed in the gen- H=γ¯h X I·A˜i·Si. (1) eral form [21]: i∈n.n. 1 γ2k χ ”(q,ω) Here γ is the gyromagnetic ratio, I is the nuclear spin = B lim F2(q) ab (2) of the In(1), Si is the spin on the ith Ce, and the sum T1T 2 ω→0Xq,α α ω is over the four nearest neighbor Ce sites. Note that whereαissummedoverthetwodirectionsperpendicular there should also exist a direct hyperfine coupling be- to the applied static field. Here F2(q) are form factors, tween the In nuclei and the conduction electrons. How- α ever, the fact that T−1 drops by almost two orders of χα(q,ω) is the dynamical susceptibility, and the sum is 1 over the Brillouin zone. The form factors are the spatial magnitude belowT suggeststhat this coupling is small N Fourier transforms of the coupling in Eq. (1), and are compared to the coupling between the nuclei and the given by: Ce moments. For notational simplicity, we define A ab caannd sxhoaws tAh|a|,t⊥K=c =Aa4bA(1cχ±c xa)n.d KUasbing=E4qA.abχ(a1b); ounse- Fa2b(q) = 16A2abcos2(cid:16)qx2a(cid:17)cos2(cid:16)qy2a(cid:17) (3) ing the slopes of Kα versus χα given above, we have 2 2 2 qxa 2 qya Aab = 4.9kOe/µB and Ac = 6.6kOe/µB. The pa- +16Aabx sin (cid:16) 2 (cid:17)sin (cid:16) 2 (cid:17) rameter x can be determined from the internal field q a q a F2(q) = 16A2cos2 x cos2 y . (4) in the antiferromagnetic state. The moments are lo- c c (cid:16) 2 (cid:17) (cid:16) 2 (cid:17) calized on the Ce sites, with the structure given by: S(ri) = S0cos(πx/a)cos(πy/a){cos(q0z),sin(q0z),0}, Since Fa2b(q) >0 for all q, T1−1 will pick up fluctuations where S0 = 0.74µB, and q0 = 0.2972cπ [16, 17, 18, 19]. atallwavevectors. In particular,the In(1) is sensitive to For this structure, the internal field is given by: H = the critical slowing down of the spin fluctuations above int 2AabxS0{sin(q0z),cos(q0z),0}. The measured internal TN. 3 We now can determine quantitatively the low fre- 5 quency spin spectrum of CeRhIn5. As may be seen in 4 Fig. (2), T1T varies quadratically with T for T > 20K, 3 whichiswhatmighthavebeenanticipatedforacollection 2 of independent Ce moments that are weakly coupled to the conduction electrons via an interaction H = Jσ·S. 1 To see this we note that the susceptibility of the local moments is given by: TB0 56 100(T χ (ω)= χ0(T) , (5) Γ/k 4 80 1AT) L 3 60 F 1−iω/Γ(T) -1 2 40 (se where χ0(T) is the bulk susceptibility (∼ 1/T for T > 20 c-1 20K), Γ(T)=πJ2N2(0)k T, and N(0) is the electronic K B 0.1 0 -1 density ofstates atE . FromEqs. (2,3,5)wethen have: 0 10 20 30 ) F 6 T (K) (T1T)−1 =4γ2kBA2ab(1+x2)χ0(T)/Γ(T)∼T−2. (6) 5 2 3 4 56 2 3 4 56 2 3 0.1 1 10 Fig. (2c)showsΓ(T)asa function oftemperature. Note T/T 0 that Γ ∼ T for T > 20K, with a slope determined by JN(0)∼0.24. AlthoughINSdataofΓ(T)isunavailable FIG.3: Γ(T)/kBT0 versusT/T0. (◦)CeRhIn5 withT0 =5K, in this material, Bao and coworkershave estimated that and (solid squares) CeCoIn with T = 17K. The solid line 5 0 Γ<<3meV for T <7K [7]. is from Cox [8]. The inset shows the build-up of antiferro- We interpret the crossover at 20K to an almost con- magnetic correlations below ∼30K follows the 1/T behavior stantvalueofΓasreflectingtheonsetofKondoscreening (solid line) described in the text, and extend to Tc = 1.8K, the transition temperature for the 5T magnetic field used in of the local moments. The relaxation of independently this experiment. screened local moments, a ”Kondo gas”, has been cal- culated by Cox et al. [22], who find Γ(T) = T0f(T/T0) where f is a universal function reflecting a temperature dependent effective coupling, and T0 is proportional to behavior, then (T1T)−AF1 ∼ ξ(T). Fig. (2b) shows the Kondo temperature. For T > T0, Qachaou et al. (T1T)−1 −(T1T)−01 versus T, where (T1T)0 = 0.00897 have shown that this scaling behavior holds for several sec K. The solid line is a fit to ξ ∼ (T −TN)−ν, where mixed-valent and heavy electron materials [23]. As may ν = 0.30 ± 0.05. As seen in Fig. (2b), the tempera- beseeninFigs. (2c)and(3),agoodfittotheexperimen- ture dependence of the NMR measurements agrees well tal data may be obtained with T0 = 5K over the range with INS measurements of ξ. Note that for a single crit- 8K Hc2 temperature dependence will be the same and the con- thatshowthespecificheatincreasesatlowtemperatures clusions will not change. as though there were an ordering temperature at T ≈ 0 Note that in both materials the data scale with the [2]. The temperature dependence of (T1T)−AF1 is signifi- Cox formula for T > T∗ ≈ 1.5T0, however below this cant. In the CeRhIn5, (T1T)−AF1,3D ∼ ξ, as expected for temperature Γ(T) is less than the Cox prediction. As 3D fluctuations. On the other hand, for quasi-2D fluc- was the case for CeRhIn5, Γ(T) drops markedly below tuations (T1T)−AF1,2D ∼ ωs−f1 ∼ ξ2. In fact, the data in the Cox ”Kondo gas” prediction below T∗, a change we CeCoIn5 are best understood in a picture of 2D fluctu- canidentifywiththeonsetofa”Kondoliquid”regimein ations, where ω ∼ T, which is exactly the result ob- sf which the interactions between the local moments gives servedin the cupratesabovethe pseudogaptemperature rise to a new physical regime. We suggest that just as [28]. In light of the behavior observed in CeRhIn5 such in the Rh material, T∗ in the Co material also marks correlationsprovideanaturalexplanationofthesuppres- the onset of antiferromagnetic correlations, and proceed sionofΓ,as wellasa possible mechanismforthe d-wave to analyze the data in CeCoIn5 below T∗ in the same superconductivity [29]. Indirect evidence for such corre- fashion as in CeRhIn5. lations has been seen in [14]. Measurements of Knight The inset of Fig. (3) shows (T1T)−AF1 = (T1T)−1 − shiftandT1−1 underpressureandasafunctionofdoping (T1T)−01 versusT,where we have taken(T1T)0 =0.0793 shouldproveinvaluabletounderstandingtheevolutionof sec K, the value at 30K. The solid line is (T −T )−1, these antiferromagnetic correlations as the ground state N with T = 0. Importantly, this result suggests that in changes from antiferromagnetic to superconducting. N the absence of superconductivity CeCoIn5 would order magnetically atT ∼0, so that in this materialone has a We thank P. C. Hammel, C. P. Slichter, W. Bao, A. quantum critical point that is obscured by superconduc- Balatsky, C. Varma, J. Lawrence, Z. Fisk, and Y. Bang tivity[26]. Quantumcriticalityisalsoconsistentwiththe for useful discussions. This work was performed under non-Fermi liquid behavior seen in the normal state [27]. the auspices of the U.S. Department of Energy. [1] H.Hegger,C.Petrovic,E.G.Moshopoulou,M.F.Hund- [17] W. Bao, et al., Phys. Rev.B 63, 219901 (2001). ley,J.L.Sarrao,Z.Fisk,andJ.D.Thompson,Phys.Rev. [18] A. Llobet (2002), private communication. Lett.84, 4986 (2000). [19] N.J.Curro,P.C.Hammel, P.G.Pagliuso, J.L.Sarrao, [2] C. Petrovic, P. G. Pagliuso, M. F. Hundley, J. D. Thompson, and Z. Fisk, Phys. Rev. B 62, R6100 R. Movshovich, J. L. Sarrao, J. D. Thompson, Z. Fisk, (2000). andP.Monthoux,JournalOfPhysics-CondensedMatter [20] et al., Eur. Phys.B 18, 601 (2000). 13, L337 (2001). [21] T. Moriya, J. Phys. Soc. Jpn. 18, 516 (1963). [3] N.D.Mathur,F.M.Grosche,S.R.Julian,I.R.Walker, [22] E.Kim,M.Makivic,andD.L.Cox,Phys.Rev.Lett.75, D. M. Freye, R. K. W. Haselwimmer, and G. G. Lon- 2015 (1995). zarich, Nature394, 39 (1998). [23] A. Qachaou, et al., J. Mag. Magn. Mat. 63-64, 635 [4] A. V. Chubukov, D. Pines, and J. Schmalian, cond- (1987). mat/0201140. [24] A. Cornelius, P. Pagliuso, M. Hundley, and J. Sarrao, [5] Y.Bang,I.Martin,andA.V.Balatsky,Phys.Rev.B66 Phys. Rev.B 6414(14), 4411 (2001). 224502 (2002). [25] In principle, one can extract x by measuring the [6] K. Miyake, S. Schmitt-Rink, and C. Varma, Phys. Rev. anisotropyofT forT >T∗.UsingEqs.(2-3,6) (T1−1)ab = B 34(9), 6554 (1986). 1 (T1−1)c [7] W. Bao, et al., Phys. Rev.B 65, 100505 (2002). 1 + A2c χc(T) . We find this ratio goes to 1/2 for [8] D. Cox, N. Bickers, and J. Wilkins, J. Appl. Phys. 57, 2 2A2ab(1+x2)χab(T) 3166 (1985). T →0, reflecting thefact thatAc vanishes. However,no [9] G. Zheng, et al., Phys. Rev.Lett. 86, 4664 (2001). physically reasonable values of x exist that satisfy the [10] Y.Kohori, et al., Phys. Rev.B 64, 134526 (2001). data for T >> 50K, which probably signifies that Γ(T) [11] T. Takeuchi, et al., J. Phys.Soc. Jpn. 70, 877 (2001). is anisotropic. [12] N.J. Curro, et al., Phys.Rev.B 64, 180514 (2001). [26] V. Sidorov, M. Nicklas, P. Pagliuso, J. Sarrao, Y. Bang, A. Balatsky, and J. Thompson, Phys. Rev. Lett. [13] P.G.Pagliuso,N.O.Moreno,N.J.Curro,J.D.Thomp- 8915(15), 7004 (2002). son, M. F. Hundley,L. L. Sarrao, and Z. Fisk (2002), to [27] A. Bianchi, et al., Phys. Rev. Lett. 8913(13), 7002 appear in Physica B. [14] S.Nakatsuji, et al., Phys. Rev.Lett. 89, 106402 (2002). (2002). [28] R. Corey, et al., Phys.Rev.B 53(9), 5907 (1996). [15] A. Christianson, A. Lacerda, M. Hundley, P. Pagliuso, and J. Sarrao, Physical ReviewB 6605(5), 4410 (2002). [29] P. Monthoux and G. G. Lonzarich, Phys. Rev. B 63, [16] W. Bao, et al., Phys. Rev.B 62, 14 621 (2001). 054529/1 (2001).

Low Frequency Spin Dynamics in the CeMIn_5 Materials

Similar Low Frequency Spin Dynamics in the CeMIn_5 Materials