Energy scales of Lu(1-x)Yb(x)Rh2Si2 by means of thermopower investigations
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b APS/123-QED
Energy scales of Lu − x Yb x Rh Si by means of thermopowerinvestigations U. K¨ohler, ∗ N. Oeschler, and F. Steglich
Max Planck Institute for Chemical Physics of Solids, D-01187 Dresden, Germany
S. Maquilon and Z. Fisk
Department of Physics and Astronomy,University of California, Irvine, CA 92697, USA (Dated: October 26, 2018)
Abstract
We present the thermopower S ( T ) and the resistivity ρ ( T ) of Lu − x Yb x Rh Si in the tempera-ture range 3 K < T <
300 K. S ( T ) is found to change from two minima for dilute systems ( x < . Si . A similar behavior has also been found for the mag-netic contribution to the resistivity ρ mag ( T ). The appearance of the low- T extrema in S ( T ) and ρ mag ( T ) is attributed to the lowering of the Kondo scale k B T K with decreasing x . The evolutionof the characteristic energy scales for both the Kondo effect and the crystal electric field splitting∆ CEF are deduced. An extrapolation of T K to x = 1 allows to estimate the Kondo temperature ofYbRh Si to 29 K. For pure YbRh Si , T K and ∆ CEF /k B lie within one order of magnitude andthus the corresponding extrema merge into one single feature. ∗ Electronic address: [email protected] . INTRODUCTION YbRh Si is a stoichiometric heavy-fermion (HF) metal with an extremely low antifer-romagnetic ordering temperature of 70 mK.[1] It crystallizes in the tetragonal ThCr Si structure and has been investigated extensively due to its pronounced non-Fermi liquid(NFL) properties at low T . This behavior was attributed to a quantum critical point, whichcan be attained by application of a small magnetic field of 60 mT within the ab plane.[2] Atintermediate temperatures (10-300 K) the properties of YbRh Si are determined by a com-petition of crystal electric field (CEF) excitations and the Kondo interaction. A knowledgeof the corresponding characteristic energy scales k B T CEF and k B T K is essential for an under-standing of the low-temperature properties of the compound. The NFL behavior dominatesthe thermodynamic and transport properties up to 10 K. Notably, the electronic specificheat divided by T exhibits a logarithmic increase upon cooling with a spin fluctuation tem-perature T = 24 K.[1] The corresponding entropy revealed a doublet ground state of theYb ions with a Kondo temperature T K of approximately 17 K.[3] The CEF level schemeof YbRh Si was determined from inelastic neutron scattering: The 4 f multiplet of theYb is split into 4 doublets at energies corresponding to 0-200-290-500 K, respectively.[4]The strong interplay between Kondo effect and CEF splitting manifests itself in thetransport properties of the system. The temperature dependencies of both resistivity [5] ρ ( T ) and thermopower [6] S ( T ) exhibit a single large extremum around 100 K, which wasattributed to scattering on the full Yb multiplet. No signature corresponding to Kondoscattering on the ground-state doublet at T K has been found in S ( T ) and ρ ( T ). Suchbehavior was observed and theoretically predicted for compounds, where the energy scales k B T CEF and k B T K are of the same order of magnitude, i.e. for systems near the crossoverfrom the HF to the intermediate-valent (IV) regime.[7, 8, 9]Upon applying pressure, the Kondo temperature of Yb-based HF systems is typicallyshifted to lower T , while the CEF levels are not affected significantly. For sufficiently smallvalues of T K separate maxima in ρ ( T ) due to Kondo effect on the ground state and onexcited CEF levels are expected to occur. Such behavior was confirmed for YbRh Si bymeans of resistivity investigations under pressure:[5] Above 4 GPa, the single peak in ρ ( T )splits into three separate maxima. The two maxima at lower T were attributed to Kondoscattering on the ground-state doublet with the onset of coherence and to Kondo scattering2n thermally populated CEF levels. The origin of a third maximum remains unclear.According to theoretical models, the lowering of T K is also expected to induce system-atic changes in S ( T ) (Ref. 8 and references therein): For systems with a thermopower asdescribed for YbRh Si the anticipated behavior upon application of a small pressure orweak lowering of T K is the appearance of a low- T shoulder. With further decreasing T K two separate minima develop in S ( T ), similar to the behavior of ρ ( T ). The minimumat lower T , which reflects Kondo scattering on the ground-state doublet, is situated at T min1 ≈ T K .[8, 10, 11] The high- T minimum is caused by Kondo scattering on thermallypopulated CEF levels. For an excited CEF level at ∆ CEF above the ground state it typicallyappears at T min2 ≈ (0 . . . . . CEF /k B .[8, 12, 13] Such evolution has been detected e.g. forYb(Ni x Cu − x ) Si upon substitution (chemical pressure).[14] The IV system YbCu Si ex-hibits a single minimum in S ( T ). As the Kondo temperature is reduced by Ni substitutionon the Cu site, a shoulder appears at low T already for the lowest Ni content studied( x = 0 . x < . T shoulder is directly related to the development of the HF state.YbRh Si seems to be an appropriate system to address this problem, since it, being a HFmetal, exhibits a single minimum in S ( T ).In order to change the effective coupling between the 4 f and the conduction electrons,substitution on all crystallographic sites has been realized in YbRh Si , generally withthe aim of lowering the antiferromagnetic ordering temperature. This, however, is con-nected with an increase in T K as observed in La − x Yb x Rh Si ,[15] I-type YbIr Si ,[16] andYbRh (Si − y Ge y ) .[17] Recent investigations on Lu − x Yb x Rh Si indicate a weak loweringof T K upon replacement of Yb by the nonmagnetic Lu.[18] Additionally, the substitutionon the Yb site leaves, in a first approximation, the chemical environment of the remain-ing Yb ions unchanged, thus reducing the influence of disorder on the magnetic moments.Lu − x Yb x Rh Si therefore appeared a promising candidate for the observation of distinctanomalies in the thermopower due to Kondo interaction on the ground state and on excitedCEF levels at ambient pressure, with the objective to determine the characteristic energyscales k B T K and k B T CEF of pure YbRh Si .In this paper we present thermopower and resistivity measurements of a number ofLu − x Yb x Rh Si single crystals with Yb concentration 0 ≤ x ≤
1. The energy scales of3ondo interaction and CEF excitations as a function of the Yb concentration are deducedbased on the thermopower and supported by the resistivity data. Finally, the evolution of S ( T ) and ρ ( T ) is discussed in comparison to other Ce and Yb-based materials. II. EXPERIMENTAL DETAILS
Single crystals of Lu − x Yb x Rh Si (0 ≤ x <
1) were grown from In flux, as describedelsewhere.[18] The average (nominal) Yb content x nom of each batch was determined fromsusceptibility measurements and confirmed by microprobe analysis on selected crystals.However, our resistivity measurements suggest a moderate variation of the Yb concentrationthroughout a batch. The scaling analysis described below indicates deviations from x nom ofup to approximately 5 %. Specifically, the samples with x nom = 0 .
15 and x nom = 0 .
49 studiedin S ( T ) and ρ ( T ) have an effective Yb concentration of x = 0 .
10 and x = 0 .
44, respectively.For all other samples the nominal concentration has been confirmed, i.e. x = x nom . In thefollowing, the effective values, x , are used.The lattice constants of the stoichiometric systems LuRh Si and YbRh Si were deter-mined from X-ray diffraction measurements on powdered material. The ThCr Si crystalstructure has been confirmed within the doping series. No additional peaks have been re-solved in the pattern. The fraction of foreign phases can thus be excluded to be higherthan 2 %. The microprobe analysis has indicated no free elemental nor binary phases inthe studied samples. Due to the extremely small change of the unit cell volume V uc of0 . ± .
12 % with respect to YbRh Si a linear dependence of V uc ( x ) is assumed for thecrystals with partial substitution. The tiny variation in the lattice constant is not expectedto significantly influence the relative position of the CEF levels.Investigations of the thermopower S and the electrical resistivity ρ were performed withinthe ab plane of the crystals with a typical size of 4 × × .
05 mm . Both quantities weremeasured in the temperature range from 3 K to 300 K in a commercial device (PPMS fromQuantum Design) using the same contacts. Measurements of ρ were extended down to 0.4K using a He insert. The resistivity was determined with a 4-point ac technique. Forthe thermopower a relaxation-time method with a low-frequency square-wave heat pulseutilizing two thermometers was used. The determination of S implies an average over thecontact area of both, the voltage and the temperature gradient. In our setup, due to the4mall crystal size, the contact size was not negligible compared to the sample dimensions.However, data sets obtained from repeated measurements on the same specimen but withdifferent contacts can be scaled on top of each other. In particular, the position of theminimum in S ( T ) remained unaffected. The absolute values of S could be reproducedwithin ± III. RESULTS
The thermopower S ( T ) of Lu − x Yb x Rh Si (0 ≤ x ≤
1) is plotted semi-logarithmically inFig. 1a ( x = 0 .
08 not shown for the sake of clarity). Fig. 1b displays the low- T behavior ofthe same curves on a linear temperature scale. The thermopower of the reference compoundLuRh Si is smaller than 1 µ V/K in this T range and therefore omitted.LuRh Si exhibits a small positive thermopower, which is typical for normal metals withhole-like charge carriers. The thermopower comprises mostly a diffusion part of light non-magnetic charge carriers. A strong phonon drag contribution leading to an enhancementaround 20 K has not been resolved. By contrast, the thermopower of YbRh Si is negativein the whole temperature range 3 K ≤ T ≤
300 K with large absolute values. It showsa single broad minimum around 80 K, as typically found in valence-fluctuating Yb com-pounds like YbCu Si .[14] However, for the HF system YbRh Si the observed behavior wasattributed to a combination of Kondo interaction and CEF effects.[6] With decreasing Ybconcentration the temperature dependence of the thermopower changes qualitatively. Thesamples with x = 0 .
75 and x = 0 .
62 exhibit a minimum at 80 K and a shoulder at low tem-peratures, which may be seen on a linear scale (Fig. 1b). For Yb concentrations of x ≤ . T upon further decreasing x . Simultaneously theabsolute values of the minimum structure at elevated temperatures are significantly reduced.For samples with x ≤ .
44 a sign change in S ( T ) appears below room temperature, whichis shifted to lower T with decreasing Yb concentration. This indicates a stronger relativeinfluence of the nonmagnetic contribution to the thermopower in these samples.The electrical resistivity of Lu − x Yb x Rh Si was measured on the same samples as thethermopower. In addition, a specimen with x = 0 .
02 has been investigated, which was too5
10 100 300-60-40-20020 x = 0 0.10 0.62 0.23 0.75 0.44 1.00 S ( V / K ) T (K)Lu Yb x Rh Si (a)0 10 20 30-40-30-20-100 x =0.10 0.62 0.23 0.75 0.44 1.00 S ( V / K ) T (K)(b)
FIG. 1: (a) Temperature dependence of the thermopower of Lu − x Yb x Rh Si (0 ≤ x ≤ S ( T ) of the crystals with x > small to measure S ( T ). The results of ρ ( T ) are in agreement with Ref. 18. The resistivityof LuRh Si takes a value of about 30 µ Ωcm at 300 K and decreases linearly from roomtemperature to 10 K, below which it reaches a constant value of 1.25 µ Ωcm. The mag-netic contribution, ρ mag , was calculated by subtracting the data of the reference compoundLuRh Si and a sample-dependent disorder term ρ . The results normalized to the Ybconcentration are shown in Fig. 2. For most samples scaling of the data at elevated temper-atures was achieved by using the value of the nominal concentration. As already mentionedabove, adjustment of the effective concentration was necessary for x nom = 0 .
15 to x = 0 . x nom = 0 .
49 to x = 0 .
44 to ensure scaling of the high- T data above 100 K.The magnetic resistivity ρ mag ( T ) of the series reflects the evolution from a diluted toa dense Kondo system, as e.g. demonstrated in Ce x La − x Cu .[19] At temperatures T >
100 K, ρ mag of all samples increases logarithmically with decreasing T . Subsequently, samples6 .3 1 10 100 300050100150 0.020.080.10*0.23 0.44*0.620.751.00Lu Yb x Rh Si m ag / Y b ( c m / Y b ) T (K) x = FIG. 2: Temperature dependence of the magnetic contribution to the resistivity of Lu − x Yb x Rh Si (0 < x ≤ ⋆ ) the effective Yb concentration deviates fromthe nominal value, namely x = 0 .
44 ( x nom = 0 .
49) and x = 0 .
10 ( x nom = 0 . with low Yb concentrations x ≤ .
23 exhibit a plateau around 60 to 100 K, followed by afurther increase in ρ mag to lower T . At T < x ≥ .
62 pass through maxima around 70 to 100 K and then drop towards lowertemperatures. The plateau or maximum at elevated T is attributed to the presence ofa CEF splitting in the system. The depopulation of excited levels upon cooling and theassociated lowering of the scattering rate leads to a reduction of ρ mag in this temperaturerange. Towards lower T the differences between the diluted and the dense Yb systemsbecome evident. While Kondo scattering on the ground-state doublet gives rise to a secondincrease as − ln T and a saturation at lowest T for x ≤ .
23, the onset of coherence promotesa further decrease in the magnetic resistivity in samples with x ≥ .
62. The sample with Ybconcentration x = 0 .
44 is situated close to the crossover between the two regimes. It showsan only weak decrease in ρ mag towards low T and a saturation at a relatively large residualvalue. IV. DISCUSSION
For a quantitative analysis of the thermopower the magnetic contribution, S mag , is usuallydetermined by use of the Gorter-Nordheim rule: S mag ρ mag = Sρ − S ref ρ ref . S ref and ρ ref are7enerally taken as the thermopower and (total) resistivity of the nonmagnetic referencecompound. The resistivity ρ ref may be approximated by a sum of a phononic contribution ρ phref and a residual resistivity ρ due to impurities. It is assumed that ρ phref does not changesignificantly upon chemical substitution. The disorder induced by doping, however, affectsthe residual resistivity, and ρ has to be replaced by the x dependent disorder contribution ρ of the alloys. Since ρ ( x ) cannot be determined accurately, an exact evaluation of S mag is almost impossible. For the presented data, the overall behavior of S mag , and especiallythe position of the shoulders is not expected to strongly deviate from that of S . At lowtemperatures ( T <
50 K), at which the thermopower and the resistivity of LuRh Si aresmall, the difference between S and S mag is negligible. Just below room temperature, thecalculation of S mag mainly implies a correction for the diffusion thermopower of light chargecarriers. The features below 100K remain basically unchanged. For the discussion of thedata we therefore analyze S ≈ S mag .Considering the Kondo and CEF energy scales of pure YbRh Si , the two shoulders ob-served in the thermopower of Lu − x Yb x Rh Si ( x < .
5) are attributed to Kondo scatteringon the ground-state doublet and on thermally populated multiplet states. Thus, the corre-sponding characteristic temperatures T S low and T S high allow an estimation of the evolution of T K and ∆ CEF upon substitution. The position of the low- T minimum in the thermopoweris usually close to the Kondo temperature of the CEF ground state,[20] in agreement withtheoretical calculations for s = 1 / T S low = T S K . The high-temperatureminimum is attributed to Kondo scattering from thermally populated CEF levels. Thus, itrepresents a characteristic temperature for the splitting of the relevant levels. Fig. 3 showsthe evolution of T S low and T S high upon substitution. They have been determined from a fit tothe data between 4 and 200 K to a sum of two negative Gaussian curves on a logarithmictemperature scale.Fig. 3a displays T S high vs. effective Yb content including the position of the single largeminimum T S min in the thermopower of pure YbRh Si . In view of the substantial uncertaintyin the determination of T S high , the position of the high- T shoulder or minimum in S ( T ) maybe regarded as temperature-independent. The mean value of 86 ±
10 K allows an estimate ofthe energetic position of the relevant excited CEF levels. As theoretical calculations predict8 .0 0.5 1.00102030 0.8 0.6 0.4 0.2 0.00.0 0.5 1.06080100 Lu Yb x Rh Si (b) S T K (x) T K ( K ) x T K (p)p (GPa) S T h i gh ( K ) xLu Yb x Rh Si (a) FIG. 3: Characteristic temperatures of Lu − x Yb x Rh Si at ambient pressure (a,b) and of YbRh Si under pressure (b). The x-axes in (b) are scaled to the same unit-cell volume. The characteristictemperatures for S ( T ) were obtained from a fit to the data as explained in the text. The errorbars represent the uncertainty of the fitting procedure. The uncertainty in the unit-cell volume,and consequently the pressure axis in (b) is denoted by an error bar on T ρ K ( p ). The solid line in(b) is a linear fit to T S low ( x ). T S high = (0 . . . . . CEF /k B ,[8, 12, 13] the observed minimum corresponds to a splitting of150-290 K with respect to the ground-state doublet. Thus, good agreement is obtained withresults of inelastic neutron scattering, which revealed doublets at 0-200-290-500 K.[4] Wetherefore conclude that the large thermopower minimum around 80 K is caused by Kondoscattering on the full Yb multiplet.Fig. 3b shows the position of the low- T minimum T S low = T S K of the samples with x < . x . For higher Yb concentrations a determination of T S low was not possible withsatisfactory precision. The data sets for low x clearly reveal an increasing T S K with risingYb concentration. A linear extrapolation of T S low ( x ) yields a Kondo temperature of 29 K for9bRh Si . The fit is shown as a line in Fig. 3b. This value is about a factor 1.5 larger thanthe one obtained from the entropy of the system, namely T c P K = 17 K.[3] However, the modelused in Ref. 3 for calculating T K based on the entropy [21] yields a temperature dependenceof c P , which deviates significantly from that of the NFL compound YbRh Si . Its applicationto this system is therefore somewhat questionable. Furthermore, a determination of T K fromdifferent experimental probes usually yields different values, however, always of the sameorder of magnitude.The Lu-Yb substitution leads to a change in the unit cell volume V uc . In order to evaluatethe relevance of this effect for the observed change in T K a comparison to results fromexperiments under pressure p is of interest. Using the bulk modulus [22] of YbRh Si of 189 GPa and assuming a linear relation of V uc vs. x we can compare our results withinvestigations under pressure. The Kondo temperature T ρ K ( p ) of YbRh Si determined fromresistive pressure studies [23] is shown as a dashed line in Fig. 3b, where the pressure axis isscaled to the same unit-cell volume change. It is seen that the lowering of T K under pressureis somewhat weaker than upon substitution. This reflects the relevance of additional effectsbeside the variation of V uc . Chemical substitution induces a change in the band structureof the system, which may influence the effective coupling between the 4 f and conductionelectrons even at constant V uc . However, the relatively small difference between the evolutionof T K under pressure and upon Lu substitution underlines a dominating influence of the unitcell volume in Lu − x Yb x Rh Si . This is ascribed to the unchanged chemical environment ofthe 4 f moments as a result of the substitution on the rare-earth ion site. Furthermore, due tothe small radius of the 4 f shell, Lu and Yb behave chemically very similar. The 4 f electronstogether with the nucleus act as an ”effective nucleus”. Therefore, Lu-Yb substitution isexpected to have a minor influence on the band structure. Yet, the integer valence v ofthe Lu ions compared to the slightly reduced value for the Yb ions (Lu vs. Yb v + with2 . ≤ v ≤ . f systems have been related tothe likewise enhanced electronic contributions to the specific heat. In the zero temperaturelimit, the ratio S/γT of several correlated compounds takes a quasi-universal value.[25] Formetals the dimensionless quantity q = N AV eS/γT with the Avogadro number N AV and theelectron charge e is close to ±
1, whereas the sign depends on the type of charge carriers.This relation can be derived within Fermi-liquid theory assuming impurity scattering as the10
10 100 300-50-40-30-20-100 1020304050202530354045Lu Yb x Rh Si S ( V / K ) T (K)x = 0.62 m ag ( c m ) -40-30-20-10010 S ( V / K ) x = 0.44 m ag ( c m ) FIG. 4: Comparison between thermopower and magnetic contribution to the resistivity for two Ybconcentrations x = 0 .
44 and x = 0 . relevant scattering process.[26] In the present system a thermopower S ∝ T is only foundfor T < x < .
5, due to both the low Kondo temperature of the order of 10 Kand the NFL behavior of the pure system YbRh Si . The calculated q values ranging from − .
63 to − .
85 are in line with that expected for hole-like charge carriers of − S ( T ) from a single minimum at x = 1 to a double-peakstructure for small x is correlated to the lowering of T K upon decreasing Yb concentration.YbRh Si with a Kondo temperature of approximately 20 K and a first excited CEF levelaround 200 K seems to be situated near the critical ratio T CEF /T K where the two minima ofKondo scattering on ground state and thermally populated CEF levels in S ( T ) merge intoa single feature. A slight reduction of T K on the other hand allows for a separation of botheffects.A similar behavior is found in the magnetic contribution to the resistivity ρ mag ( T ) ofLu − x Yb x Rh Si . The similarities are best seen for the two samples with Yb concentrationsat the crossover from a single large minimum in S ( T ) to two clearly separated shoulders.Fig. 4 shows a comparative plot of S ( T ) as well as ρ mag ( T ), which were determined on11he same specimens with x = 0 .
44 and 0.62. The curves for thermopower and magneticresistivity strongly resemble each other. For the sample with x = 0 .
44 two clearly separatedshoulders are seen in both quantities, situated at the same temperatures as indicated bythe vertical lines. By contrast, the sample with x = 0 .
62 exhibits only one large extremumin S ( T ) and ρ mag ( T ) around 80 K. The double-peak structure is well tracked by both thethermopower and resistivity for x < .
5, whereas larger concentrations x > . x > . S ( T ) and ρ ( T ),although the corresponding energy scales differ by approximately one order of magnitude.This observation can be understood in view of the relatively large temperature range, whichis effected by the thermal population of an excited CEF level as apparent e.g. from a Schottkycontribution to the specific heat. A significant population of an excited CEF level and,consequently, an enhanced scattering from CEF excitations sets in at temperatures wellbelow the splitting of ∆ CEF /k B . Thus, in Lu − x Yb x Rh Si for x > . S ( T ) and ρ ( T ) is caused by scattering onthe full Yb multiplet. On the other hand, in samples with x < . T , whilethermal population of the three excited levels give rise to a second broad extremum atelevated temperatures.The disappearance of the low- T minimum in the thermopower of Yb(Ni x Cu − x ) Si for x = 0 is connected with a crossover to the valence-fluctuating regime.[14] A qualitativelysimilar behavior has been frequently observed in Ce systems under pressure and upon sub-stitution, e.g. in CeRu Ge [27] or Ce(Ni x Pd − x ) Si .[28] In contrast to Yb compounds, Cebased HF systems generally exhibit maxima in S ( T ) due to the preponderance of electron-like charge carriers. Likewise, the crossover from two maxima to a single maximum is usuallytaken as an indication [27, 28] that T K lies in the order of the CEF splitting and the systementers the valence-fluctuating regime. However, in XANES measurements of the L-III ab-sorption edge of the Yb ion in Lu − x Yb x Rh Si no significant contribution from Yb couldbe resolved at 5 K. Taking into account the resolution of the method, the valence v has beenestimated to be 2 . ≤ v ≤ . − x Yb x Rh Si appears to be a rare example, for whichthe two extrema in S ( T ) merge, while the system is still in the HF regime. The coalescenceof both features corresponding to T K and ∆ CEF can be observed in detail in this series sincethe Lu substitution induces an extremely small change of the unit-cell volume connectedwith a very weak lowering of T K . V. SUMMARY
The temperature dependence of the thermopower of Lu − x Yb x Rh Si qualitativelychanges upon substitution from a single large minimum in S ( T ) for the pure Yb compoundto two well separated minima for x < .
5. A similar evolution is found in the magneticcontribution to the resistivity. This change in the overall behavior of S ( T ) and ρ mag ( T )is ascribed to a lowering of T K upon decreasing Yb content. For high Yb concentrations x > . T K due to the substitution of Ybby Lu on the other hand allows for a separation of both effects in the transport propertiesof the series for x < .
5. The evolution of the Kondo temperature upon substitution canbe understood mainly from the change in the unit-cell volume. In addition, modificationsin the band structure may be relevant. Due to the extremely small overall change in T K ,Lu − x Yb x Rh Si displays the crossover from two minima in S ( T ) to one minimum uponincreasing x without entering the valence-fluctuating regime. The Kondo temperature ofYbRh Si has been estimated to be around 29 K. Acknowledgments
We are grateful to V. Zlati´c, S. Burdin, and C. Geibel for stimulating discussions. Wethank N. Caroca-Canales for X-ray diffraction measurements on LuRh Si . UK acknowl-edges financial support by COST action P16. Work at UC Irvine (SM and ZF) has been13upported by NSF Grant No. DMR-0710492. [1] O. Trovarelli, C. Geibel, S. Mederle, C. Langhammer, F. M. Grosche, P. Gegenwart, M. Lang,G. Sparn, and F. Steglich, Phys. Rev. Lett. , 626 (2000).[2] P. Gegenwart, J. Custers, C. Geibel, K. Neumaier, T. Tayama, K. Tenya, O. Trovarelli, andF. Steglich, Phys. Rev. Lett. , 056402 (2002).[3] J. Ferstl, Ph.D. thesis, TU Dresden, 2007.[4] O. Stockert, M. M. Koza, J. Ferstl, A. P. Murani, C. Geibel, and F. Steglich, Physica B , 157 (2006).[5] G. Dionicio, H. Wilhelm, G. Sparn, J. Ferstl, C. Geibel, and F. Steglich, Physica B ,50 (2005).[6] S. Hartmann, U. K¨ohler, N. Oeschler, S. Paschen, C. Krellner, C. Geibel, and F. Steglich,Physica B , 70 (2006).[7] K. Alami-Yadri, D. Jaccard, and D. Andreica, J. Low Temp. Phys. , 135 (1999).[8] V. Zlati´c and R. Monnier, Phys. Rev. B , 165109 (2005).[9] Y. Lassailly, A. K. Bhattacharjee, and B. Coqblin, Phys. Rev. B , 7424 (1985).[10] N. E. Bickers, D. L. Cox, and J. W. Wilkins, Phys. Rev. B , 2036 (1987).[11] G. D. Mahan, Phys. Rev. B , 11833 (1997).[12] A. K. Bhattacharjee and B. Coqblin, Phys. Rev. B , 3441 (1976).[13] S. Maekawa, S. Kashiba, M. Tachiki, and S. Takahashi, J. Phys. Soc. Jap. , 3194 (1986).[14] D. Andreica, K. Alami-Yadri, D. Jaccard, A. Amato and D. Schenck, Physica B ,144 (1999).[15] J. Ferstl, C. Geibel, F. Weickert, P. Gegenwart, T. Radu, T. L¨uhmann, and F. Steglich,Physica B , 26 (2005).[16] Z. Hossain, C. Geibel, F. Weickert, T. Radu, Y. Tokiwa, H. Jeevan, P. Gegenwart, and F.Steglich, Phys. Rev. B , 094411 (2005).[17] S. Mederle, R. Borth, C. Geibel, F. M. Grosche, G. Sparn, O. Trovarelli, and F. Steglich,J. Phys.: Condens. Matter , 10731 (2002).[18] S. Maquilon, Ph.D. thesis, UC Davis (unpublished), 2007.[19] Y. ¯Onuki and T. Komatsubara, J. Magn. Magn. Mater. , 281 (1987).
20] D. Huo, J. Sakurai, T. Kuwai, T. Mizushima, and Y. Isikawa, J. Appl. Phys. , 7634 (2001).[21] H.-U. Desgranges and K. D. Schotte, Phys. Lett. A , 240 (1982).[22] J. Plessel, M. M. Abd-Elmeguid, J. P. Sanchez, G. Knebel, C. Geibel, O. Trovarelli, and F.Steglich, Phys. Rev. B , 180403(R) (2003).[23] G. A. Dionicio, Ph.D. thesis, TU Dresden, 2006.[24] U. Burkhard et al. (unpublished).[25] K. Behnia, D. Jaccard, and J. Flouquet, J. Phys.: Condens. Matter , 5187 (2004).[26] K. Miyake and H. Kohno, J. Phys. Soc. Jpn. , 254 (2005).[27] H. Wilhelm and D. Jaccard, Phys. Rev. B , 214408 (2004).[28] D. Huo, J. Sakurai, O. Maruyama, T. Kuwai, and Y. Isikawa, J. Magn. Magn. Mater. ,202 (2001).,202 (2001).