Quantum critical behavior in the heavy Fermion single crystal Ce(Ni 0.935 Pd 0.065 ) 2 Ge 2
C. H. Wang, J. M. Lawrence, A. D. Christianson, S. Chang, K. Gofryk, E. D. Bauer, F. Ronning, J. D. Thompson, K. J. McClellan, J. A. Rodriguez-Rivera, J. W. Lynn
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Quantum critical behavior in the heavy Fermion single crystal Ce(Ni . Pd . ) Ge C. H. Wang , , , J. M. Lawrence , A. D. Christianson , S. Chang , K. Gofryk , E. D. Bauer ,F. Ronning , J. D. Thompson , K. J. McClellan , J. A. Rodriguez-Rivera , , J. W. Lynn University of California, Irvine, California 92697 Los Alamos National Laboratory,Los Alamos, NM 87545 Neutron Scattering Science Division,Oak Ridge National Laboratory, Oak Ridge, TN, 37831 NCNR, National Institute of Standards and Technology,Gaithersburg, MD 20899-6102 Department of Materials Science and Engineering,University of Maryland, College Park, MD 20742 ∗ We have performed magnetic susceptibility, specific heat, resistivity, and inelastic neutron scatter-ing measurements on a single crystal of the heavy Fermion compound Ce(Ni . Pd . ) Ge , whichis believed to be close to a quantum critical point (QCP) at T = 0. At lowest temperature(1.8-3.5K), the magnetic susceptibility behaves as χ ( T ) − χ (0) ∝ T − / with χ (0) = 0 . × − m /mole(0.0025 emu/mole). For T <
C/T = γ − T / with γ of order 700 mJ/mole-K . The resistivity behaves as ρ = ρ + AT / for temperatures below 2K. This low temperature behavior for γ ( T ) and ρ ( T ) is in accord with the SCR theory of Moriyaand Takimoto[1]. The inelastic neutron scattering spectra show a broad peak near 1.5 meV thatappears to be independent of Q ; we interpret this as Kondo scattering with T K = 17 K. In addition,the scattering is enhanced near Q =(1/2, 1/2, 0) with maximum scattering at ∆ E = 0.45 meV; weinterpret this as scattering from antiferromagnetic fluctuations near the antiferromagnetic QCP. In strongly correlated electron systems, a quantum critical point (QCP) separates an antiferromagnetic (AFM) orferromagnetic(FM) state from a nonmagnetic Fermi liquid state at T = 0 K. This QCP can be tuned by adjusting acontrol parameter such as doping parameter x , external pressure P , or applied magnetic field H . In the vicinity ofthe QCP, the critical fluctuations are quantum in nature and induce unique behavior. The nature of this quantumground state phase transition poses one of the most significant challenges in condensed matter physics. Heavy Fermion(HF) compounds are very good candidates for studying the QCP. These compounds behave as Fermi Liquids (FL),with large values for the specific heat coefficient γ = C/T and susceptibility χ (0), and with the resistivity varying as∆ ρ ∝ T at low temperature. When such systems are tuned close to a QCP, fluctuations of the nearby magneticallyordered state affect the thermodynamic behavior and lead to Non-Fermi Liquid (NFL) behavior such as ∆ ρ ∝ T α with α < C/T ∝ ln ( T /T ) at low temperature[2].The approach to a QCP has been attained in HF systems by doping in the tetragonal compounds Ce(Ru − x Rh x ) Si (x=0.03)[3, 4] and Ce − x La x Ru Si (x=0.075)[5, 6] as well as in the orthorhombic compound CeCu − x Au x (x=0.1)[7–9], by application of a magnetic field in YbRh Si [10], and with no additional control parameter in the compound β -YbAlB [11].A conventional spin fluctuation theory has been established to explain the non-Fermi liquid behavior. This theory,implemented through a renormalization-group approach[12, 13] or through the self-consistent renormalization(SCR)method[1], has successfully explained various aspects of the non-Fermi liquid behavior. However, the experimentalresults for some systems do not follow this spin fluctuation theory. A locally critical phase transition has been invokedto explain the behavior of a few systems[14]. Hence, measurements in other systems are needed to understand thebehavior near the QCP in HF materials.Below 2 K, the tetragonal compound CeNi Ge exhibits behavior for the resistivity and specific heat that obeysthe predictions of the SCR theory for a 3D AFM QCP[1]: ∆ ρ ∝ T / and C/T ∝ γ (0) − A √ T [15, 16]. Inelasticneutron scattering shows two low energy features[17–19]. A broad peak at 4 meV that is only weakly Q -dependentcorresponds to Kondo scattering with T K = 46 K[20]. A peak at 0.7 meV that is highly Q -dependent and showsa maximum intensity at Q N = (1/2 1/2 0) corresponds to scattering from antiferromagnetic fluctuations. Althoughthis compound is clearly close to a QCP, when T decreases below 0.4 K or B increases above 2 T, the system entersthe FL state and C/T shows saturation[15, 16, 21]. The compound can be brought closer to the QCP by alloyingwith Pd. According to the phase diagram proposed by Knebel et al[22], the QCP occurs in Ce(Ni − x Pd x ) Ge when ∗ Electronic address: [email protected] x =0.065. Fukuhara et al found[23] that when x = 0.10, the ordering wavevector remains Q N = (1/2 1/2 0); for larger x it changes to Q N = (1/2 1/2 1/6).Here we report measurements of the magnetic susceptibility, the specific heat for a series of magnetic fields, theresistivity, and the inelastic neutron scattering of a single crystal of Ce(Ni . Pd . ) Ge grown with 58-Ni to reducethe incoherent scattering. For this alloy, we report the inelastic neutron spectra for the first time.Single crystals were grown using the Czochralski method. The magnetization was measured in a commercial su-perconducting quantum interference device (SQUID) magnetometer. The specific heat measurements were performedin a commercial physical properties measurement system (PPMS). The electrical resistivity was also measured in thePPMS using the four wire method. The inelastic neutron experiment were performed on the MACS spectrometer[24]at the NIST center for neutron research (NCNR). ( e m u / m o l e ) Ce(Ni Pd ) Ge ( - m / m o l e ) T (K) / ( m o l e / e m u ) / ( m o l e / m ) T (K) - (0) ~ T -1/6 (0)=0.032 X 10 -6 m /mole =0.0025 emu/mole 0.0030.0060.0090.01280859095100 FIG. 1: (a): Magnetic susceptibil-ity of Ce(Ni . Pd . ) Ge . Thesolid line is the high temperatureCurie-Weiss fit. (b): Low tempera-ture inverse susceptibility. The solidline is χ ( T ) − χ (0) ∝ T − / . Ce(Ni Pd ) Ge B=0T B=3T B=6T B=9T C / T ( m J / m o l e - K ) (a) T (K) C / T ( m J / m o l e - K ) (c) =695.19 mJ/mol-K a=263.23mJ/mol-K C/T= -a T C / T ( m J / m o l e - K ) (b) FIG. 2: (a): C/T in applied fieldsof B=0, 3 T, 6 T and 9 T. (b): Lowtemperature zero field C/T curvein a logarithmic temperature scale.(c) Low temperature zero field C/Tdata. The solid line is γ − aT / . ( c m ) =a+bTa=2.26 cmb=0.24 cm/KCe(Ni Pd ) Ge (a) T(K) ( c m ) T (K) =A+BT
A=2.32 cmB=0.14 cm/K (b) =a+bT
FIG. 3: Resistivity forCe(Ni . Pd . ) Ge . Thesolid line is the linear fit for thetemperature range 0.4 K to 12 Kwhile the dashed line represents T / dependence in the temperaturerange 0.4 K to 2 K. The magnetic susceptibility χ ( T ) is shown in figure 1(a). At high temperatures the data follow Curie-Weiss (CW)behavior χ ( T ) = C eff / ( T − θ ); in the range 150-300 K, the CW fit yields a moment 2.84 µ B and θ = -118 K. Below100 K, χ ( T ) is enhanced over the CW value but no long range magnetic order is observed down to 1.8 K. In thetraditional spin fluctuation theory[1, 2, 12, 13] for a three dimensional system near a QCP, χ ( T ) ∝ T − / is expectedfor an AFM while χ ( T ) ∝ T − / is expected for a FM. In figure 1 (b), our data for Ce(Ni . Pd . ) Ge followthe behavior of ( χ ( T ) − χ (0)) ∝ T − / . Exponents for the susceptibility that are smaller than 1 are also observed for β -YbAlB [11] where χ ( T ) ∝ T − / when B=0.05 T and for YbRh (Si . Ge . ) where ( χ ( T ) − χ (0)) ∝ T − . [25].In figure 2(a) we plot the specific heat C/T . The zero field data are very similar to that reported by Kuwai et al forCe(Ni . Pd . ) Ge [26]. The data are logarithmic with temperature in the range 1 K to 5 K. When an externalfield is applied, the data deviate from the ln T behavior and show saturation. As for CeNi Ge , this is because themagnetic field forces the system to enter the FL state. However, in figure 2(b), the low temperature ( T < T behavior; in this temperature range the data can be fit(figure 2(c)) to theform γ − aT / (with γ of order 695 mJ/mole-K ), which as mentioned is the expected behavior in the SCR theory.In figure 3(a) the resistivity ρ ( T ) data are seen to be roughly linear in temperature over a wide range 0.4 K to12 K, indicating NFL behavior. On the expanded scale of figure 3(b), ρ ( T ) deviates from T -linear, following a T / dependence. Again, this is the expected behavior in the SCR theory.Hence, at temperatures higher than 2 K, this alloy exhibits typical NFL behavior, with the resistivity varying as ρ ( T ) ∝ T , the specific heat varying as C/T ∝ ln T T , and the susceptibility varying as ( χ ( T ) − χ (0)) ∝ T − / . Attemperatures below 1 K, however, ρ ( T ) ∝ T / and C/T ∝ γ − a √ T , thereby following the expected behavior of theSCR theory for a three dimensional AFM QCP spin fluctuation system.In figure 4, we plot the INS spectra for Ce(Ni . Pd . ) Ge . A broad peak near 1.5 meV appears to beonly weakly Q -dependent. As for the 4 meV peak in CeNi Ge , we identify this as representing Kondo scattering. E (meV) I n t en s i t y ( a r b . un i t s ) I(1/2 1/2 0)-I(3/4 3/4 0)(b)
Ce(Ni Pd ) Ge , T=0.4 K Q=(1/2 1/2 0) Q=(3/4 3/4 0) I n t en s i t y ( a r b . un i t s ) E (meV)(a)
FIG. 4: Inelastic neutron scattering spec-tra for Ce(Ni . Pd . ) Ge . The datawere collected on MACS. In (a), the scat-tering at the critical wavevector Q N =(1/2 1/2 0) is compared to the scatteringat (3/4 3/4 0). The lines represent fits tothe sum of a quasi-elastic and an inelasticLorentzian peak, as described in the text.In (b) the difference between the data for Q = (1/2 1/2 0) and (3/4 3/4 0) is com-pared to a quasielastic Lorentzian (solidline). Extra intensity is observed when ∆
E < Q = (1/2 1/2 0). This is the wavevector of theantiferromagnetic fluctuations seen for CeNi Ge near 0.7 meV, hence we identify it as the critical wavevector Q N for the QCP.We first fit these spectra to the sum of a quasi-elastic Lorentzian (peak width Γ ) and an inelastic Lorentzian (peakposition E and width Γ ) (Fig. 4(a)). For Q = (1/2 1/2 0) we find E =1.51 meV, Γ =0.90 meV and Γ =0.35 meV;for Q =(3/4 3/4 0), we find E =1.35 meV and Γ =0.88 meV. In figure 4(b), we use a second approach to obtain theAFM fluctuation spectra: we subtract the spectra for Q = (3/4 3/4 0) from that measured at (1/2 1/2 0). A fit ofthe difference to a quasielastic Lorentzian peak gives the AFM fluctuation energy scale as Γ = 0.44 meV.The value of E is around 1.35-1.51 meV represents a Kondo temperature in the range 15.7-17.5 K. We note thatthis is smaller than the value 46 K seen in CeNi Ge , so that the approach to the QCP involves a reduction of T K .We note in addition that T K does not vanish at the QCP. A finite Kondo temperature is expected for a spin densitywave system at a QCP.The value of the AFM fluctuation energy Γ is around 0.35-0.44 meV appears to be smaller than the fluctuationenergy scale of pure CeNi Ge where Γ=0.7 meV [17]. This represents ”critical slowing down” of the AFM spinfluctutaions. However, despite the fact that this system is believed to sit at the QCP, the lifetime of the AFMspin fluctuations is finite, i.e. it does not diverge as expected at the critical point. Such a ”saturation” of the spinfluctuation lifetime has been observed for other HF systems where the QCP is attained by alloying [4, 6] and mayarise from the fact that the QCP occurs in a disordered environment.Research at UC Irvine was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Divisionof Materials Sciences and Engineering under Award DE-FG02-03ER46036. Work at Los Alamos National Laboratorywas performed under the auspices of the U.S. DOE/Office of Science. Work at ORNL was sponsored by the LaboratoryDirected Research and Development Program of ORNL, managed by UT-Battelle, LLC, for the U. S. DOE, and wassupported by the Scientific User Facilities Division, Office of Basic Energy Sciences, DOE. We acknowledge thesupport of the National Institute of Standards and Technology, U. S. Department of Commerce, in providing theneutron research facilities used in this work. The identification of commercial products does not imply endorsementor recommendation by the National Institute of Standards and Technology. [1] Moriya T and Takimoto T 1995 J. Phys. Soc. Jpn. Rev. Mod. Phys. J. Phys. Soc. Jpn Phys. Rev.
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