Non-Fermi-liquid and multichannel Kondo phenomena in Y1−xUxPd3 and related alloys

PHYSICA ELSEVIER

Physica B 199&200 (1994) 396-401

Non-Fermi-liquid and multichannel Kondo phenomena in Yl-xUxPda and related alloys C.L. Seaman ~'*, M.B. Maple ~'b ~Department q/ Physics and brstitute /or Pure and Applied Phy:~ical Sciences, Universi O, 0["Cahfornh~--San Diego. La Jolla, CA 92093, USA bCenter.for Materials Science, Los Alamos National Laboratory, Los Alamos, NM 87545. USA

Abstract A brief review of experimental results which reveal non-Fermi-liquid behavior in the low-tonperature properties of the Mt-xU.,Pda (M = rare earth or Th) Kondo alloy system is presented. Measurements of lattice parameters, electrical resistivity, magnetic susceptibility, low-temperature specific heat, and magnetic neutron scattering are compared to predictions of the multichannel Kondo model, and are generally consistent with a two-channel spin-l/2 quadrupolar Kondo effect. Possible deviations from the model due to crystalline electric field effects, interionic interactions, disorder, and Fermi-level tuning aspects are discussed.

The alloy Y~-xUxPd3 is a complex yet fascinating system, exhibiting many interesting phenomena including a structural change of phase, unusual spin-glass behavior, crystalline electric field (CEF) effects, Fermi-level tuning, and unusual Kondo behavior Il]. These phenomena affect the measurable properties to varying deg"ees which can make their interpretation difficult. For low U concentrations, Kondo behavior is observed, but with unusual low-temperature properties characterized by non-Fermi-liquid behavior [l, 2]. This observation has generated much interest and a large amount of experimental work. in this paper, we review the expertmental observations, with an emphasis on the low-temperature non-Fermi-liquid behavior observed for low U concentrations, where the unus,.al Kondo behavior dominates. In seeking a microscopic explanation, we compare the experimental results to the multichannel Kondo model [3] and find general agreemem [1] with * Corresponding author.

a two-channel quadrupolar Kondo effect, which arises from an antiferromagnetic exchange interaction between the electric quadrupole moment of the !"3 ground state doublet of the U 4 + ions and pseudospins associated with two time-reversed conduction band channels [4]. Alternative explanations include interactions between impurity moment~,, which increase with increasing impurity concentration I-2], and the presence of disorder for sufficiently dirty metals [5]. In our most recent work, we have attempted to confirm or refute the quadrupolar Kondo effect as a microscopic mechanism, primarily through investigations m the more clllute L.] impurity limit and establishrr~ent ofa F 3 CEF ground state of the U 4. ions. We find thz~t the non-Fermi-liquid behavlor persists for lower U concentrations, where interimp,..,ty interactions are less important. Our results also support the existence of a I-3 ground state doublet which is split, presumably from local disorder, which in turn might account for deviations of the data from thec~ry at the lowest temperatures.

0921-4526/94/$07.00 t:~ 1994 Elsevier Science B.V. All rights reservcd SS[,I 0921-4526(93) E0362-K

C.L. Seaman. M.B. Maple / Physica B 199&200 (1994) 396-401

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Fig. 1. (a) Cubic lattice parameters of M~ ~U~Pd3 compounds as a function of x. Solid lines represent linear fits to the data, which extrapolate (dashed lines} to a common value of 4.10 + 0.005 ~ for cubic UPd3. This value is hypothetical since UPd 3 forms in the hexagonal NiaTi crystal structure. (b} Lattice parameters of cubic M Pd 3 compounds versus ionic radius of the trivalent M ion. Filled circles are taken from (a]: open circles are for RPd 3 (R = rare earth) from Ref. [7] and have been multiplied by a factor of 1.003. With the exception of LaPd3, the data fall on a straight line. The hypothetical value for cubic UPd3 obtained from ~a) !filled squares) is consistent with U being tetravalent in the M~_ ,U~Pd3 compounds.

As an extension of our original work on the ~-xUxPd3 system, we have made similar investigations of the M~ -xUxPd3 series of compounds (M = Sc, Y. La, and Pr) [_6]. Powder X-ray diffraction patterns for these trivalent M ions reveal that M~_xU~Pd 3 forms in the Cu3Au cubic crystal structure for x _< 0.5. Samples with x = 0.6 (M = Y and Sc) showed mixed cubic and hexagonal phases. For higher x _> 0.9 (M = Y), samples are again single phase, but form the Ni3Ti hexagonal structure. This change in phase is most likely electronically driven by the tetravalence of the U ions. Figur',- I(a) shows the cubic lattice parameters obtained from X-ray diffraction patterns for M~_,~UxPd 3 as a function of x.

397

Least-squares linear fits are shown as solid lines which extrapolate (dashed lines) to a common value of 4.100 + 0.005/~ for cubic UPda. This value is hypothetical since Ur',-t ~ is hexagonal: attempts to form the cubic phase ha,, far been unsuccessful. Nevertheless, this value is co .~red in Fig. l(b) to the lattice parameters of cubic MPda compounds, which when plotted as a function of M 3 + ionic radius are seen to fall on a straight line. The largest of these, M = La, is an exception. Data are taken from Fig. l(a) for x = 0 (filled circles) and from Ref. [7] for various RPd3 (R = rare earth) compounds (open circles), whose lattice parameter values have been multiplied by a factor of 1.003 in order to be consistent with our data. This small discrepancy is most likely due to systematic errors in one or both X-ray studies. Comparing the hypothetical value of 4 . 1 0 0 + 0 . 0 0 5 A for cubic UPd3 (filled squares), we see that this value is consistent with U being tetravalent in M~_.,U.~Pd~. In addition, the observation of Fermi-level tuning in Y1-xU~Pd3, as well as the absence of Fermi-level tuning in hexagonal MI-~U~Pd3 (M *+ = Zr and Th), implied that U is tetravalent in all of these Mi-~,UxPd3 compounds. Fermi-level tuning refers to the increase of the Fermi energy EF as tetravalent ions (in this case U) are substituted for trivalent Y, first observed in photoemission spectroscopy measurements on Yl_~UxPd 3 [8], which causes the magnitude of the 5f binding energy [E~f - Evl to increase. For a Kondo alloy, larger binding energies lead to smaller Kondo temperatures TK, in agreement with our observation of a decreasing TK with x for Y~ _.,UxPd 3, described be!ow, and with y in the system Y o , ~.Th~Uf~1Pd3 [9]. The existence of a Kondo effect is evident in the Y~ -~UxPd3 system for x _< 0.3, in which the temperature dependence of the electrical resistivity p(T}, shown in Fig. 2 for 0.02 _< x _< 0.2, increases logarithmically at high temperatures T > T~ The U contribution to the resistivity Ap(T} was estimated by subtracting p ( T ) of the host YPd3 compound. Estimates of the Kondo temperature TK as the temperature at which the normalized resistivity Ap{ T)/Ap(O) equals 0.8, given by the horizontal line in Fig. 2, yield TK values which increase with increasing x, in accordance with Fermi-level tuning, from ---50K for x = 0 . 2 to ---200K for x = 0 . 0 2 . At low temperatures ( T < 7k), p ( 7 ) does not saturate quadratically as ApI T)/Ap(O) = I - [ T/aTK]",

ill

with n = 2, as one expects for a standard S = !/2, singlechannel [m = 1} Kondo effect, or more generally for m < 2S, which reflects the behavior of a Fermi liquid. Rather, we find n ~ !.0 +_ 0.2 for all samples in the temperature range T < 20 K. In this temperature range, p~ T}

C,L, Seaman, M,B. Map~e~ Physica B 199&200 (1994) 396-401

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Fig. 2. Ca) Normalized electrical resistivity Ap(T)/Ap(O) versus log T for Yt-xU~Pd~ (0.02 _< x < 0.2) and Yo.sTho.tUo.~Pd3 revealing Kondo behavior with Kondo temperature TK, defined as the temperature at which Ap(T)/Ap(O) = 0.8, which increases with decreasing x due to Fermi-level tuning. Diluting Y,.sUo.2Pda with Th does not affect TK very much.

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,,_ of YPda is temperature independent; therefore the exponent n does not depend on the subtraction of the host contribution to p(T), although the coefficient (aT}:) will, especially for samples with small x. Taking a value for a of 4.3, values of TK from these power-law fits vary from ,-- 40 K for x = 0.2 to --- 500 K for x = 0.05. It is difficult to determine the power-law temperature dependence o f p ( T ) for smaller values of x because of the large values of TK which yield small d p ( T ) / d T slopes at low temperatures. In an attempt to a p p r o a c h the single impurit, y limit while holding TK a p p r o x i m a t e l y constant, we have m a d e similar p ( T ) measurements on the Yo.sTho.2-xU.~Pd3 (0.01 _< x < 0.2) series of alloys, shown in Fig. 3. Because nonmagnetic Th is tetravalent, the total concentration of tetravalent ions remains fixed at 0.2, which should yield a constant value TK ~ 40 K according to the Ferm!-Ievel tuning scenario, all else remaining constant. F r o m a similar analysis of p(T), TK a p p e a r s to increase slightly with decreasing x. The most significant result, however, is that the low-temperature power-law fits to Eq. (1}, shown in Fig. 31b), again yield exponents of n = 1.0 + 0.1 in the temperature range T < 20 K. The electrical resistivity thus has a non-Fermi-liquid linear temperature dependence at the lowest measured temperatures even for samples with U concentrations as low as 1%. It therefore seems unlikely that this feature is due to interimpurity interactions. Measurements to lower temperatures are planned. Measurements of p ( T ) for the Sc~ _ .~UxPd3 system also yield values of n ~ 1 [6].

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Fig. 3. (a) Normalized electrical resistivity Ap(T)/Ap(O) versus log T for Yo.sTho.: .,U.~Pd3 (9.01 < x < 0.2} revealing Kondo behavior with Kondo temperature TK, which increases only slightly with decreasing x. (b): Low-temperature resistivity data (T < 20 K) plotted as Iog~l - AplT)/Ap(O)) versus log T. Solid lines represent fits to a power law Ap(T)/Ap(O) = I - (T/aTK)", " with n = 1.0 _ 0.1 for all samples, exhibiting non-Fermi-liquid behavior. This value is in between that expected for singlechannel (n = 2) and two-channel 01 = 1/2) S = 1/2 Kondo models.

A recent exact solution of the multichannel K o n d o model by c,~,~r,~rm.~l s,.la ,h,-,-,ry rl01 ¢,-,,,,a ,t,,~t ,, ll'~ for the m = 2, S = I/2 case. in the T = 0 asymptotic limit, which a p p e a r s to be in disagreement with our data. Measurements to lower temperatures would therefore be interesting. It m a y be possible that further refinement of the theory to describe p{ T) at intermediate temperatures T < T• will yield a larger value ofn. O n e possible reasc, n for the discrepancy between experiment and theory is a splitting of the F~ ground doublet as explained below. Figure 4 shows the impurity contribution to the specilic heat AC, plotted as A C / T versus log 7", for several

C.L. Seaman, M.B. Maple/Physica B 190&200 11994) 396 401 .4

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~amples which display a characteristic logarithmic divergence over the temperature range 0.7 K < T < 10 K. Such non-fermi-liquid behavior is in striking contrast to the S = 1/2, m = 1 standard K o n d o effect, where AC/T approaches a constant value which varies as 1/TK at low temperatures (T,~ irK). At higher temperatures, the data increase sharply due to incorrect subtraction of the phonon contribution and/or a Schottky anomaly due to an excited state CEF level above the ground ~tate. A lowtemperature logarithmic divergence is theoretically predicted for the S = I/2, ,:: = 2 K o n d o model, described by the equation A C / T = - (0.251/TK) InIT/0.41TK) [11]. The solid lines in Fig. 4 represent fits of the data to this equation plus a background term b. We extract TK from the slope of these lines, and b from the offset, values for which are given in Table 1. Values for b are comparable to measured 7 values for the host metals, ;' ~ 3.5, 6.9, and 8.2 mJ/mol K 2 for YPd3 [12], Yo.sTho.2Pd3, and ScPd3 [12], respectively, which have not been subtracted from the data. A hybridization-broadened Schottky anomaiy at higher temperatures can also contribute to h. Especially noteworthy is the nearb identicai behavior of the more dilute compound Yo.sTho.lUo.lPd3 to that of Yo.sUo.2Pd3, which suggests that the observed nonFermi-liquid behavior is not due to interimpurity interactions, which decrease with decreasing U concentration, but is dominated by single impurity effects. Parameter values for Sco. 7Uo.3Pd 3 are also comparable to Yo.sUo.zPd3, as seen in Table 1. Because of the smaller size of Sc compared to Y, which gives rise to stronger

hybridization and/or a larger density of states at the Fermi energy N(EF) of the host metal, Scl _.,U:Pd~ has a larger TK value than Yt -xUxPd3 for given x [6]. Fits of the AC/T data for Yo.gUo.~Pd3 yield a larger TK value than for Yo.sUo.zPd3, in agreement with the resistivity data and with the Fermi-level tuning model. A remarkable aspect of the data is that the associated entropy S(T) far all of these samples saturates to a value close to ~'R/2)ln 2 before continuing to increase due to the highertemperature upturns in AC/T seen in Fig. 4 [1]. This suggests a finite zero-temperature entropy of the same value, in order that the full degeneracy of the ground doublet be recovered at high temperatures. Such an unnsual T = 0 entropy is predicted for the S = 1/2, m = 2 Kondo model [10, !1]. There is a noticeable deviation of the AC/Tdata from the fits at low temperatures (T~<0.7 K) where the data increase more strongly with decreasing T. Possible causes include a splitting of the F3 ground state, evidence for which is described below, and residual spin-glass-like freezing at these low temperatures in a small fraction of the sample [13]. Figure 5 shows the normalized magnetoresistance plH}/plO) of Yo.sUo.2Pd 3 at 0.6 K in applied magnetic fields H up to 30T. Data for 7.5 T _< H _< 30T were obtained at the Francis Bitter National Magnet Laboratory at MIT. The magnetoresistance has a quadratic field dependence for low H < 6 T and can be described by the equation plHj/p(O)= 1-(H/Ho) 2, with Ho ~ 80T. At 30 T, pIH)/p(O) drops by only ~ 5%. For a magnetic Kondo effect, H acts to lift the degeneracy of the (hybridized) iocai impurity state, polarizing tile relative o~:cupation of the levels. In very large magnetic fields tldl >>ki~T~), the low-temperature (T<-.{ T~:) magnetoresistance approaches zero when complete polarization is achieved. Comparing our data to the J = 1:2 Coqblin Schrieffer model, for example, a 5% drop at 30 T, which corresponds to a magnetic energy of -,- 20 K, implies a fairly large value of TK "- 170 K. For an electric quadrupolar IS = 1'2. m = 2) Kondo effect. H lifts the chamwl degeneracy, which causes the system to approach single-channel q,]adrupolar Kondo behavior ia which

C.L. Seaman. M.B. Maple/ Physica B 199&200 (1994) 396-401

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the low-temperature resisSvity remains large and finite in large fields (#BH >>kRT~). Cox [14] has estimated that p(H)/p(O) should drop to a value of no less than 0.9, which is not inconsistent with the observed value of 0.95 at 30 T ~ 20 K ~ TK/2. In addition, the temperature dependence of the resistivity p(T) will approach single-channel Kondo behavior, which is that of a Fermi liquid with a quadratic temperature dependence at low temperatures (T <{ TKt. An increase in the low-temperature power-law exponent n from Eq. (1) has been observed by us up to 6 T [1] and by Andraka and Tsvelik up to 14 T [2]. In our investigations of the M~ -xUxPd3 (M = Sc, Y, La, Pr) system, Kondo behavior is observed for M = Sc, Y, and Pr, at low U concentrations, with a TK scale that decreases with increasing size of the M 3 + ion, for given x. This can be understood in terms of a ciecrease in the hybridization strength and/or the host density of states at the Fermi energy N(EF) with expansion of the lattice, b~th of which serve to decrease TK I-6]. For M = La, no K o n d o effect is observed in p(T) for any x down to at least 1.2 K due to the large size of La 3+. The lAtt ~I I~Pd,, series therefore serves ~ :, reference .... -','~ in which the magnetic behavior of the U ions can be studied in the absence of a Kondo effect, even at low U concentrations. Figure 6 shows the temperature dependence of the low-temperature specific heat AC of L a o , U o t Pd3 after subtraction of an estimated phonon contribution lIT 3, where/~ = 0.71 mJ/mol K 4 was determined from a hightemperature fit of the date,. The data display an anomaly which peaks at T = 5 K (2.4 K) when plotted as A(" (AC'T) versus T, and has an associated entropy of R In 2, implying a doublet ground state. One interprctation of

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Fig. 6. Low-temperature specific heat AC (solid circles} versus temperature Tof Lao.gUoaPd3, which does not exhibit a Kondo effect in the electrical resistivity down to 1.2 K. The solid line represents a fit to a two-level Schottky anomaly resulting from a Gaussian distribution of level splittings centered at 0.8 meV with a FWHM of 1.6 meV. The temperature derivative of the electrical resistivity dp/dT is shown as the dashed line. The similar temperature dependence to AC(T) suggests that both anomalies arise from temperature-dependent depopulation of CEF levels.

this result is that the nonmagnetic !"3 doublet ground state is split due to local deviation from cubic symmetry at the U site. The solid line in Fig. 6 represents a fit ofthe data to the Schottky anomaly which results from a Gaussian distribution of such level splittings centered about a value AcEr = 0.8 meV ( ,-, 9 KJ with a full width at half maximum (FWHM) of 1.6 meV. The dashed line in Fig. 6 shows the temperature derivative of p(T). The similar temperature dependence to AC(T) supports this interpretation. The low-temperature magnetic susceptmility of this compound points to a nonmagnetic ground state, implying that the 1"3 doublet lies the lowest. No irreversibility was observed in a small field of H = i Oe, down to 1.8 K. However, a Yo.¢Tho.sUo.~Pd3 sample, which has a peak in AC(T) that is nearly identica! to that for the Lao ,Uo. t Pd.~ compound, did reveal a slight irreversibilit3', indicating spin-glass freezing, below a temperature Tso ~- 2.4 K. This temperature coincides with the peak in AC(T)/T [9]. A possible scenario is that a Jahn-Teiler distortion occurs in these samples, induced by quadrupolar interactions, which lifts the degeneracy of the ground F3 doublet. A small magnetic moment could also be induced on the U ions by this distortion, producing spin-glass behavior, through a mixing of the F3 ground state with the excited CEF state which results in a nonzero expectation value of the magnetic moment of the

C.L. Seaman, M.B. Maple/ Physica B 199&200 (1994) 396 401 perturbed ground state. Indeed, the behavior of this system is reminiscent to that of nonstoichiometric PrPl-.~, an induced-moment spin-glass system [15]. Furthermore, recent muon spin relaxation measurements of the Y~ -xUxPd3 system demonstrate that the spin-glass order in this system can be understood in the context of induced m o m e n t s [13]. F o r an electric quadrupolar K o n d o effect, a splitting of the Fa doublet is analogous to the application of a m a g netic field H for a magnetic two-channel K o n d o effect which splits the impurity ground-state doublet by an energy A. In either case, the effect is a crossover to Fermi-liquid behavior for kaT < d2/kaTK when ,42 ~ ksT~, and to marginal Fermi-liquid behavior for A2/kBTK ~, kBT ~, knT~ [14]. If we take A ~ 0.8 meV 9 K from the specific heat data, this corresponds to /4 ~ 14 T for S = 1/2; for Yo.sUo.2Pd3 with TK ~, 40 K, A/kBTK ~ 0.2. Such a value could yield power-law behavior of p(T), given by Eq. (1) with an exponent n > 1/2. In addition, the specific heat AC/T is theoretically predicted for the two-channel K o n d o effect to increase below a temperature Tthat depends on the ratio A/kaTe, due to the development of a low-temperature Schottky-like anomaly from the splitting [16]. For A/kBT~ ~, 0.2, this increase occurs below T ~ 0.1 TK ~, 4 K. A somewhat smaller splitting A might account for the upturn in the AC/Tdata seen for T~<0,7 K. Finally, we remark that recent inelastic neutron scattering (INS) measurements of Yo.sUo.zPd 3 at low temperatures [17] reveal only extremely weak quasi-elastic scattering with a width less than 0.2 meV, a value much less than the Kondo temperature for this system and implying a nonmagnetic ground state. In addition, two broad magnetic peaks are observed in the INS spectrum at energies -,- 6 and -,- 16 meV, which are attributed to hybridization-broadened C E F levels. The relative integrated intensities of these peaks are consistent with F~ and F,, excited state triplets, resp,,ctively, above a F 3 ground state. It is clear that further investigations are required before this fascinating system is completely understood. We thank our many collaborators for their contributions to this work. Research at U C S D was supported by the National Science Foundation under G r a n t NL~.

401

DMR-91-07698 and by the US Department of Energy under G r a n t No. DE-FGO3-86ER45230. Work at Los Alamos was performed under the auspices of the US Department of Energy. Some equipment used in this research was provided by the Center for Interface and Materials Science and funded by the W.A. Keck Foundation.

References [I] C.L. Seaman, M.B. Maple, B.W. Lee, S. Ghamaty, M.S. Torikachvili, J.-S. Kang, L.Z. Liu, J.W. Allen and D.L. Cox, Phys. Rev. Lett. 67(1991)2882; J. Alloys Compounds 181 0992) 327. [2] B. Andraka and A.M. Tsvelik, Phys. Rev. Lett. 67 (1991) 2886. [3] P. Nozieres and A. Blandin, J. Phys. (Paris141 [1980) 193. [4] D.L. Cox, Phys. Rev. Lett. 59 [1987) 1240. [5] V. Dobrosavljevic, TR. Kirkpatrick and G. Kotliar, Phys. Rev. Lett. 69 (1992} I! 13. [6] D.A. Gajewski, P. Allenspach, C.L. Seaman and M.B. Maple, Physica B 199&20011994) 419. C.L. Seaman et al., to be published. [7] W.E. Gardner, J. Penfold, T.F. Smith and IR. Harris, J. Phys. F 2 [1972D 133. [8] J.-S. Kang, J.W. Allen, M.B. Maple, M.S. Torikachvili, W.P. Ellis, 88. Pate, Z.-X. Shen. J.J. Yeh and i. Lindau, Phys. Rev. B 39 (1989) 13529. [9] M.B. Maple, D.A. Gajewski, C.U Seaman and J.W. Allen, Physica B 199&200 (1994) 423. [10] A.W.W. t, udwig and I. At'fleck. Phys. Rev. Lelt. 67 ~1991~ 3160. [11] A.M. Tsvelik, J. Phys. C 18 (1985) 159; P.D. Sacramento and P. Schlottmann, Phys. Lett. A 142 (1989) 245. [12] M.J. Besnus, J.P. Kappler and A. Meyer, J. Phys. F 13 (19831 597. [13] W.D. Wu, A. Keren, UP. Le, G.M. Luke, B.J. Sternlieb, Y.J. Uemura, C.L. Seaman, Y. Dalichaouch and M.B. Maple, to be published (1993). [14] D.L. Cox, private communication. [15] S.K. Hasanain, R.P. Guertin, K. Westerholt, M. Guyot and S. Foner, Phys. Rev. B 24 (1981) 5165. [16] P.D. Sacramento and P. Schlottmann, Phys. Rev. B 43 (1991~ 13294. [17] H.A. Mook, C.I. Scaman. M.B. Maple, M.A. Lopez de la Torre, D.L. Cox and M. Makivic, Phvsica B 186-188 ~19931 341: H.A. Mook et al., to be published.