Phase diagram and thermal expansion measurements on the system of URu_{2-x}Fe_xSi_2
S. Ran, C. T. Wolowiec, I. Jeon, N. Pouse, N. Kanchanavatee, K. Huang, D. Martien, T. DaPron, D. Snow, M. Williamsen, S. Spagna, M. B. Maple
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Phase diagram and thermal expansion measurements on the system of URu − x Fe x Si S. Ran,
1, 2
C. T. Wolowiec,
1, 2
I. Jeon,
1, 2, 3
N. Pouse,
1, 2
N. Kanchanavatee,
1, 2
K. Huang,
1, 2, 3
D. Martien, T. DaPron, D. Snow, M. Williamsen, S. Spagna, and M. B. Maple
1, 2, 3 Department of Physics, University of California, San Diego, USA Center for Advanced Nanoscience, University of California, San Diego, USA Materials science and Engineering program, University of California, San Diego, USA Quantum Design, Inc., San Diego, USA (Dated: September 12, 2018)Thermal expansion, electrical resistivity, magnetization, and specific heat measurements wereperformed on URu − x Fe x Si single crystals for various values of the Fe concentration x in both thehidden order (HO) and large moment antiferromagnetic (LMAFM) regions of the phase diagram.Our results show that the paramagnetic (PM) to HO and LMAFM phase transitions are manifesteddifferently in the thermal expansion coefficient. For Fe concentrations near the boundary betweenthe HO and LMAFM phases at x c ≈ I. INTRODUCTION
The search for the order parameter of the hidden or-der (HO) phase in URu Si has attracted an enormousamount of attention for the past three decades. Thesmall antiferromagnetic moment of only ∼ µ B /Ufound in the HO phase is too small to account for the en-tropy of ∼ T = 17.5 K. A first-order transition from the HO phase to a large momentantiferromagnetic (LMAFM) phase occurs under pres-sure at a critical pressure P c that lies in the range 0.5- 1.5 GPa. Many studies suggest that the HO andLMAFM phases are intimately related and that a com-prehensive investigation of both phases will be useful inunraveling the nature of the order parameter of the HOphase. Although the order parameters are presumablydifferent in the HO and LMAFM phases, the two phasesexhibit almost indistinguishable transport and thermo-dynamic properties. This behavior has been referred toas “adiabatic continuity”. We have recently demonstrated that tuning URu Si by substitution of Fe for Ru affords the opportunity tostudy both the HO and LMAFM phases and the HO-LMAFM transition at atmospheric pressure. Specifi-cally, the substitution of the smaller Fe ions for Ru ionsin URu Si appears to act as a chemical pressure suchthat the temperature vs. Fe substitution phase diagramfor the URu − x Fe x Si system resembles the temperaturevs. applied pressure phase diagram for URu Si . In aprevious study, neutron diffraction measurements werecarried out on single crystal samples of URu − x Fe x Si forvarious values of x . The results revealed that the mag-netic moment increases abruptly to a maximum value at x = 0.2, above which it then decreases slowly with x , sup- porting the interpretation that tuning by Fe substitutionacts as a chemical pressure. Therefore, we have a uniqueopportunity to perform experiments at ambient pressurethat will allow us to study, with confidence, the LMAFMphase that is known to exist under high pressure.On the other hand, the phase boundary between theHO and LMAFM phases has not been definitively de-termined for the URu − x Fe x Si system. Extensive efforthas been expended to map out the precise phase bound-ary between the HO and LMAFM phases in URu Si under pressure, which is not vertical on the T - P phasediagram. The critical pressure at which the HO-LMAFM transition occurs is about 1.5 GPa at T , whileit drops to about 0.8 GPa at the base temperature. Therefore, at intermediate values of pressure, e.g., 1 GPa,URu Si goes through two successive phase transitionsupon cooling: a second order transition from the param-agnetic (PM) into the HO phase, and then a first ordertransition from the HO into the LMAFM phase. Thishas not been observed in either polycrystalline or sin-gle crystal samples of URu − x Fe x Si . It is possible thatin the polycrystalline samples, both the PM-HO andHO-LMAFM transitions are broadened, especially in thevicinity of the HO-LMAFM phase boundary, so that thetransition from the HO phase into the LMAFM phasewith decreasing temperature is not readily discernible.Neutron diffraction measurements have been carried outon the single crystals, but only for a few selected concen-trations. In the experiments reported herein, we performed ther-mal expansion, electrical resistivity, magnetization, andspecific heat measurements on URu − x Fe x Si single crys-tals for various values of x throughout both the HO andLMAFM regions of the phase diagram. Interestingly, thetransition from the HO to the LMAFM phase with de-creasing temperature in URu Si that occurs under pres-sure was previously observed by means of thermal expan-sion measurements. Our thermal expansion measure-ments reveal differences in the features associated withthe HO and LMAFM phase transitions that appear inthe thermal expansion coefficient, reflecting differencesin the coupling of the two phases to the lattice. For Feconcentrations near the boundary between the HO andLMAFM phases at x c ≈ II. EXPERIMENTAL METHODS
Single crystals of Fe substituted URu Si were grownby the Czochralski method in a tetra-arc furnace. Elec-trical resistivity measurements were performed using ahome-built probe in a liquid He Dewar by means ofa standard four-wire technique at 16 Hz using a Lin-ear Research LR700 ac resistance bridge. Magnetizationmeasurements were made in magnetic fields of 0.1 T us-ing a Quantum Design magnetic property measurementsystem (MPMS). Specific heat measurements were per-formed in a Quantum Design Dynacool Physical Prop-erty Measurement System (PPMS) using a heat-pulsetechnique. Thermal expansion measurements were madein a Quantum Design PPMS with a dilatommetry option.
III. RESULTS
Shown in Figure 1(a) are electrical resistivity ρ ( T )data, normalized to room temperature values, for variousURu − x Fe x Si compounds. The ρ ( T ) curves are offsetvertically for clarity. For this study, we focus on the in-terrelation of the HO and LMAFM phases. Therefore, wedid not perform low temperature measurements to studysuperconductivity. The transition from the PM into theHO phase in the parent compound is manifested as ananomaly at around 17 K. The transition temperature canbe extracted from the minimum in d ρ /d T . Upon Fe sub-stitution, the signature of the phase transition is pre-served, while the transition temperature T changes sys-tematically. After an initial suppression down to 16.2 Kat x = 0.08, T increases up to 34 K at x = 0.7. Simi-lar results are obtained from magnetization M ( T ) data,as shown in Fig. 2. The corresponding feature for thephase transition in M ( T ) is the slope change. This canbe seen more clearly from the quantity d( MT )/d T , whichis expected to yield a feature that is similar in shape to that observed in the specific heat data. Althoughthe signature of the phase transition in both ρ ( T ) and M ( T ) seems to remain unchanged throughout the entireFe concentration range sampled in this study, the neutrondiffraction experiments indicate that the ground stateof URu − x Fe x Si changes from the HO to the LMAFMphase, with a phase boundary close to x c = 0.1.
15 20 25 30 35 400.20.40.60.81.00 ( K ) T (K)
0 0.05 0.08 0.1 0.12 0.15 0.2 0.3 0.7 (a) URu Fe x Si (b) d [( K ) ]/ d T ( / K ) T (K)
FIG. 1. (Color online) (a) Electrical resistivity ρ and (b)derivative of ρ , d ρ /d T , vs. temperature T for URu − x Fe x Si single crystals with various values of x between 0 and 0.7.Both quantities are normalized to the room temperature val-ues. (b)URu Fe x Si
0 0.05 0.08 0.1 0.12 0.15 0.2 0.3 0.5 M / H ( - e m u / m o l O e ) T (K) (a) d ( M T / H ) / d T ( - e m u / m o l O e ) T (K)
FIG. 2. (Color online) (a) Magnetization M and (b) deriva-tive of MT , d( MT )/d T , vs. temperature T for URu − x Fe x Si single crystals with various values of x between 0 and 0.7. Displayed in Figure 3 is the electronic contribution tothe specific heat, C e ( T ), divided by temperature T , vs T , determined by subtracting the phonon contributionto the specific heat C ph ( T ) from the measured specificheat C ( T ), as discussed in our previous work. The elec-tronic specific heat C e ( T ) exhibits a well-defined BCS-like anomaly upon transition from the PM into the HOand LMAFM phases at T . The size of the jump at thetransition decreases above x = 0.2 and broadens consider-ably at x = 0.7. In our previous work on polycrystallinesamples, a broad shoulder above T was observed andattributed to disorder . There is no sign of the broadshoulder in the C e ( T ) vs. T data for the single crystalspresented here, showing that the single crystals do notsuffer from similar problems. URu Fe x Si C e / T ( J / M o l - K ) T (K)
FIG. 3. (Color online) Electronic specific heat divided by tem-perature, C e / T , vs. temperature T for URu − x Fe x Si singlecrystals with various values of x between 0 and 0.7. Presented in Figures 4(a) and (b) are the linear ther-mal expansion coefficients in the ab -plane, α ab , and alongthe c -axis, α c , vs. temperarture T for URu − x Fe x Si single crystals with various values of x between 0 and0.7. The linear thermal expansion coefficient is stronglyanisotropic, with α ab positive and α c negative. At theHO-LMAFM phase transition, an anomaly is observed inboth α ab and α c . However, the signature of the anomalyis markedly different for the PM to HO and LMAFMtransitions, showing that the HO and LMAFM phases areclearly distinct from one another. For x < T is relatively weak, while, for larger x , where the LMAFM phase is the ground state, the sizeof the jump at T is more than three times larger. Sim-ilar differences in the size of the jumps in α ab and α c for the PM-HO and PM-LMAFM transitions have beenreported for URu Si under pressure. One of the basicfeatures of URu Si is the significant amount of couplingof the HO phase to the lattice. However, the net volumechange is even larger for the PM-LMAFM phase transi-tion, as shown in Fig. 4(c), indicating that the LMAFMphase is even more strongly coupled to the lattice thanthe HO phase.It is noteworthy that for x = 0.08 and 0.1, there aretwo separate anomalies, indicating two phase transitions.Based on the size of the jump, the feature at highertemperature is consistent with the PM-HO phase transi-tion, while the feature at lower temperature is consistentwith the transition from the HO phase into the LMAFMphase. Similar results have been reported for URu Si under pressure, where the transition from the HO phaseinto the LMAFM phase was also observed in thermal ex-pansion measurements. The sign of the thermal ex-pansion coefficient indicates that as the sample is cooledfrom the HO phase into the LMAFM phase, the ab -planeshrinks and the c -axis expands, leading to an abrupt in-crease in the c / a ratio. An alternative scenario is that for x = 0.08 and 0.1, the samples are a mixture of the HO andLMAFM phase, which cannot be conclusively excluded.However, if this were the case, one would expect a smallerjump in α at the PM-HO phase transition, correspond-ing to the percentage of the HO phase. We do not see anobvious change in the size of jump for x = 0.08 and 0.1,indicating it is more likely due to two successive phasetransitions. The reason why this phase transition fromthe HO to LMAFM phase is not clearly manifested inother transport or thermodynamic measurements, e.g.,electrical resistivity, magnetization, or specific heat (asshown in Fig. 5), is not clear and requires further inves-tigation. ( - / K ) (a)ab 0 0.05 0.08 0.1 0.12 0.15 0.2 (b) (c) ( - / K ) T (K) ( - / K ) T (K)
FIG. 4. (Color online) Linear thermal expansion coefficientsin (a) the ab -plane ( α ab ) and (b) along the c -axis ( α c ) vs.temperature T for URu − x Fe x Si single crystals with variousvalues of x between 0 and 0.2. (c) The calculated volumethermal expansion coefficient β , derived from the α ab and α c vs. T data in (a) and (b), respectively. IV. DISCUSSION
Based on the electrical resistivity, magnetization, spe-cific heat and thermal expansion coefficient data, we wereable to establish the temperature T vs Fe concentration x phase diagram for the URu − x Fe x Si single crystalsshown in Fig. 6. In contrast to previous studies of poly-crystalline samples, the PM-HO phase boundary deter-mined in this study decreases slightly with increasing Fesubstitution. The LMAFM phase becomes stable at x = 0.1 where the PM-LMAFM phase boundary intersectsthe PM-HO phase boundary. For further increases in Fe
10 15 20 25 (d)(c)(b) / ( K ) (a) -0.050.000.05 dd T ( K ) ( / K ) M / H ( - e m u / m o l e O e ) d ( M T / H ) / d T ( - e m u / m o l e O e ) C e ( J / M o l - K ) ab ( - / K ) T (K)
FIG. 5. (Color online) (a) Normalized electrical resistivity ρ / ρ (295 K), (b) magnetization M/H , (c) electronic specificheat C e and (d) thermal expansion coefficient ( α ab ) vs. T fora URu − x Fe x Si single crystal with x = 0.1. Derivatives of ρ / ρ (295 K) and MT /H with respect to T vs. T are shown inred. substitution, the PM-LMAFM phase boundary increasesto 34 K by x = 0.7.This overall behavior is reminiscent of what was pre-viously observed in the dependence of the HO-LMAFMphase boundary on pressure for single crystals of the par-ent compound URu Si , indicating that the interpreta-tion of Fe substitution as a chemical pressure still ap-plies to the single crystals. The slight depression of theHO phase boundary with x for single crystal specimensmay be caused by disorder associated with the Fe substi-tution, in addition to the chemical pressure. It is clearthat for the intermediate levels of Fe concentration, x =0.8 and 0.1, there are two distinct phase transitions asobserved in the thermal expansion measurements. Thetransitions occurring at lower temperatures help iden-tify the phase boundary between the HO and LMAFMphases. This HO-LMAFM phase boundary can be ex-tended to zero temperature somewhere between x = 0.05and 0.08. It is now believed that the HO and LMAFMphases exhibit different symmetry with distinct order pa-rameters. Therefore, according to Landau theory, thereshould be a first order phase transition between the twophases with different symmetry. However, no obvioushysteresis has been observed in thermal expansion mea-surements, which indicates that the HO-LMAFM transi-tion is weakly first-order.In order to further investigate the two distinct phasetransitions detected at intermediate Fe concentrations,thermal expansion measurements were performed inmagnetic fields up to 9 T on a URu − x Fe x Si single crys- PMHO
URu Fe x Si M C p -HO -AFM T ( K ) x LMAFM
FIG. 6. (Color online) Temperature, T , vs. Fe concentration, x , phase diagram for the URu − x Fe x Si system. The phase di-agram is based on measurements of electrical resistivity, mag-netization, specific heat, and thermal expansion coefficient asa function of temperature on single crystal specimens. tal with x = 0.1. The magnetic field was applied alongthe c -axis, which is the easy-axis, and the thermal ex-pansion was measured in the same direction. The re-sults are shown in Fig. 7. Both transitions are system-atically suppressed to lower temperatures by the mag-netic field. However, the rate of suppression is clearlydifferent; the higher temperature transition is reducedonly slightly, while the lower temperature transition issuppressed more rapidly at a rate of 7.5 K/T. The be-havior of the two transition temperatures as a functionof magnetic field is shown in the inset of Fig. 7. Theseresults, together with data in high magnetic field, are consistent with our hypothesis that the transition athigher temperature is the PM-HO phase transition, whilethe one at lower temperature is the HO-LMAFM phasetransition.At the onset of the HO phase, as well as the LMAFMphase, there is a reconstruction of the Fermi surface,as revealed by transport and thermodynamic measure-ments. For the parent compound, an energy gap as-sociated with the HO phase, originally attributed to acharge or spin density wave, opens over about 40% ofthe Fermi surface. In order to determine how the energygap evolves upon Fe substitution, we have evaluated thesize of the gap by performing fits of relevant theoreticalmodels to the features observed in measurements of theelectrical resistivity, specific heat, and the thermal expan-sion coefficient that characterize the HO and LMAFMphases.Below T , the electrical resistivity ρ ( T ) consists ofcontributions from the residual resistivity, Fermi liquidelectron-electron scattering, and electron-magnon scat-tering due to spin excitations with an energy gap ∆.Since the magnons have antiferromagnetic character the ( - / K ) T (K)
0T 3T 5T 6T 7T 8T 9T c, Hllc T ( K ) H (T)
FIG. 7. (Color online) Thermal expansion coefficient alongthe c -axis vs. temperature T for a URu − x Fe x Si single crys-tal with x = 0.1 in magnetic fields up to 9 T. The magneticfield was applied along the c -axis. The magnetic field de-pendence of the two transition temperatures is shown in theinset. following expression for ρ ( T ) is appropriate: ρ ( T ) = ρ + AT + B ∆ ( T / ∆) . (1 + 2 / T / ∆)+ 2 / T / ∆) ) exp (∆ /T ) . (1)The specific heat C e ( T ), displays a well-defined, BCS-like anomaly upon transition into the HO and LMAFMphases at T . Below the transition, the C e ( T ) data canbe described by the expression: C e ( T ) = Aexp (∆ /T ) + γT, (2)where ∆ is the gap that opens over the Fermi surfaceand γ is electronic specific-heat coefficient. The vol-ume thermal expansion coefficient β ( T ), exhibits a BCS-like anomaly upon transition into the HO and LMAFMphases as well. Therefore, the β ( T ) data can be describedby a similar expression: β ( T ) = Cexp (∆ /T ) + AT. (3)The gap values extracted from the three different typesof measurements are shown in Fig. 8, together with rep-resentative fitting curves. Although the gap values aredifferent in magnitude, the three types of measurementsshow a consistent trend. The gap values for the HO phaseare relatively small, ∼
100 K from thermal expansion, ∼
80 K from specific heat, and ∼
60 K from resistance.When crossing from the HO phase to the LMAFM phase( x = 0.1 and 0.12), the gap value is suddenly enhancedby ∼
20 K. This is consistent with what has been seenfor URu Si under pressure. For second-order phase transitions, the uniaxial pres-sure derivatives of the transition temperature, dT / dP , x = 0.12x = 0.12 (f)(e) (d)(c) (b) / ( K ) fit(a)x = 0.12 ( K ) C e ( J / M o l - K ) C e fit C ( K ) C e ( - / K ) T (K) fit ( K ) x FIG. 8. (Color online) Representative fits of the expressionsgiven in the text to the (a) electrical resistivity ρ ( T ), (c) elec-tronic specific heat C e ( T ) and (e) volume thermal expansioncoefficient β ( T ) data, and values of the energy gaps ∆ ρ , ∆ C ,and ∆ β , extracted from the fits (b, d, f). can be estimated using the Ehrenfest relation, dT /dP i = V m T c ∆ α i / ∆ C p , (4)where V m is the molar volume which can be calculatedfrom lattice parameters, ∆ α i is the change in the linear( i = a , c ) or volume ( α v = β ) thermal expansion coeffi-cient at the phase transition, and ∆ C p is the change inthe specific heat at the phase transition. The inferreduniaxial pressure derivatives are shown in Fig. 9. Notethat for x = 0.08 and 0.1, both the PM-HO and PM-LMAFM phase transitions were detected in thermal ex-pansion measurements, whereas only the PM-HO phasetransition was detected in specific heat measurements.Therefore, for x = 0.08 and 0.1, we are only able to esti-mate the pressure-dependence of the transition tempera-ture for the HO phase. The uniaxial pressure derivativesof the transition temperature have different signs for thetwo crystallographic orientations. Uniaxial pressure ap-parently applied along the a - or b -axes should produce anincrease in T , whereas uniaxial pressure applied alongthe c -axis should result in a decrease in T . Anotherstriking feature is that the pressure derivatives for thetransition temperature change dramatically when cross-ing the HO-LMAFM phase boundary. For the HO phase,d T /d P i is relatively small, 0.6 K/GPa for in-plane uni-axial pressure and 0.2 K/GPa for c -axis uniaxial pressure,and does not vary much with Fe concentration. On theother hand, for the LMAFM phase, d T /d P i is signifi-cantly enhanced to above 2 K/GPa for both orientations.For hydrostatic pressure, the derivative of the transi-tion temperature with respect to pressure can be calcu-lated using the expression: dT /dP V = 2 dT /dP a + dT /dP c , (5) a c hydrostatic (calculated) measured d T / d P i ( K / G P a ) x FIG. 9. (Color online) Uniaxial pressure derivatives of thetransition temperature, T , estimated using the Ehrenfest re-lation, as well as from direct measurement. which also shows dramatic changes upon crossing thephase boundary. The hydrostatic pressure derivatives of T , estimated using the Ehrenfest relation can be quali-tatively compared with the recent results of direct mea-surements of electrical resistivity of URu − x Fe x Si underpressure, which are also shown in Fig. 9. The two setsof data match quite well for the entire range of Fe con-centration, except for x = 0.1, where the measured valueis significantly higher than the one estimated using theEhrenfest relation. As noted above, for x = 0.1, the valueof d T /d P is estimated for the HO phase only. On theother hand, under applied pressure the ground state for x = 0.1 changes from the HO phase to the LMAFM phase,indicated by a slight kink in the T ( P ) phase line. There- fore, the measured value of d T /d P is closer to that ofthe LMAFM phase. This offers an explanation for thediscrepancy between the measured and estimated valueof d T /d P at x = 0.1. V. CONCLUDING REMARKS
We have performed thermal expansion, electrical re-sistivity, magnetization, and specific heat measurementson URu − x Fe x Si single crystals for various values of x between 0 and 0.7, in both the HO and LMAFM regionsof the phase diagram. Our results show that the PM-HOand PM-LMAFM phase transitions are expressed differ-ently in the thermal expansion coefficient, indicating adifferent coupling of the two phases to the lattice. Bymeans of thermal expansion measurements, we also ob-served a possible phase transition from the HO into theLMAFM phase for intermediate levels of Fe substitution,which has not been observed in other types of measure-ments. In addition, the uniaxial pressure derivatives ofthe transition temperature, derived from thermal expan-sion and specific heat data, changes dramatically whencrossing from the HO to the LMAFM phase. VI. ACKNOWLEDGEMENT
Single crystal growth and characterization at UCSDwas supported by the US Department of Energy, Officeof Basic Energy Sciences, Division of Materials Sciencesand Engineering, under Grant No. DEFG02-04-ER46105.Low temperature measurements at UCSD were spon-sored by the National Science Foundation under GrantNo. DMR 1206553. High pressure research at UCSD wassupported by the National Nuclear Security Administra-tion under the Stewardship Science Academic AllianceProgram through DOE under Grant No. de-na0001841. T. T. M. Palstra, A. A. Menovsky, J. van den Berg, A.J. Dirkmaat, P. H. Kes, G. J. Nieuwenhuys, and J. A.Mydosh, Phys. Rev. Lett. , 2727 (1985). M. B. Maple, J. W. Chen, Y. Dalichaouch, T. Kohara, C.Rossel, M. S. Torikachvili, M. W. McElfresh, and J. D.Thompson, Phys. Rev. Lett. , 185 (1986). W. Schlabitz, J. Baumann, B. Pollit, U. Rauchschwalbe,H. M. Mayer, U. Ahlheim, and C. D. Bredl, Z. Phys. B:Condens. Matter , 171 (1986). J. A. Mydosh and P. M. Oppeneer, Rev. Mod. Phys. ,1301 (2011). C. Broholm, J. K. Kjems, W. J. L. Buyers, P. Matthews, T.T. M. Palstra, A. A. Menovsky, and J. A. Mydosh, Phys.Rev. Lett. , 1467 (1987). H. Amitsuka, M. Sato, N. Metoki, M. Yokoyama, K. Kuwa-hara, T. Sakakibara, H. Morimoto, S. Kawarazaki, Y.Miyako, and J. A. Mydosh, Phys. Rev. Lett. , 5114(1999). J. R. Jeffries, N. P. Butch, B. T. Yukich, and M. B. Maple,Phys. Rev. Lett. , 217207 (2007). G. Motoyama, N. Yokoyama, A. Sumiyama, and Y. Oda,J. Phys. Soc. Jpn. , 123710 (2008). N. P. Butch, J. R. Jeffries, S. Chi, J.B. Leo, J. W. Lynn,and M. B. Maple, Phys. Rev. B , 060408(R) (2010). F. Bourdarot, E. Hassinger, S. Raymond, D. Aoki, V. Tau-four, L.-P. Regnault, and J. Flouquet, J. Phys. Soc. Jpn. , 064719 (2010). Y. J. Jo, L. Balicas, C. Capan, K. Behnia, P. Lejay, J.Flouquet, J. A. Mydosh, and P. Schlottmann, Phys. Rev.Lett. , 166404 (2007). N. Kanchanavatee, M. Janoschek, R. E. Baumbach, J. J.Hamlin, D. A. Zocco, K. Huang, and M. B. Maple, Phys.Rev. B , 245122 (2011). P. Das, N. Kanchanavatee, J. S. Helton, K. Huang, R. E.Baumbach, E. D. Bauer, B. D. White, V. W. Burnett, M.B. Maple, J. W. Lynn, and M. Janoschek, Phys. Rev. B , 085122 (2015). F. Bourdarot, A. Bombardi, P. Burlet, M. Enderle, J. Flou-quet, P. Lejay, N. Kernavanois, V. P. Mineev, L. Paolasini,M. E. Zhito-mirsky, and B. Fk, Physica B H. Amitsuka, K. Matsuda, I. Kawasaki, K. Tenya, M.Yokoyama, C. Sekine, N. Tateiwa, T. C. Kobayashi, S.Kawarazaki, and H. Yoshizawa, J. Magn. Magn. Mater. , 214 (2007). H. Amitsuka, K. Matsuda, M. Yokoyama, I. Kawasaki,S. Takayama, Y. Ishihara, K. Tenya, N. Tateiwa, T. C.Kobayashi, and H. Yoshizawa, Physica B , 925 (2008). E. Hassinger, G. Knebel, K. Izawa, P. Lejay, B. Salce, andJ. Flouquet, Phys. Rev. B , 115117 (2008). P. G. Niklowitz, C. Pfleiderer, T. Keller, M. Vojta, Y.-K.Huang, and J. A. Mydosh, Phys. Rev. Lett. , 106406(2010). G. Motoyama, T. Nishioka, and N. K. Sato, Phys. Rev.Lett. , 166402 (2003). S. Kambe, D. Aoki, B. Salce, F. Bourdarot, D. Braithwaite,J. Flouquet, and J.-P. Brison, Phys. Rev. B , 115123(2013). M. E. Fisher, Phil. Mag. , 1731 (1962). A. de Visser, F. E. Kayzel, A. A. Menovsky, J. J. M.Franse, J. van den Berg, and G. J. Nieuwenhuys, Phys.Rev. B , 8168 (1986). D. Aoki, F. Bourdarot, E. Hassinger, G. Knebel, A. MiyakeMIYAKE, S. Raymond, V. Taufour, and J. Flouquet, J.Phys. Soc. Jpn. , 053701 (2009). S. Ran, etc. (unpublished). J. Schoenes, C. Schnenberger, J. J. M. Franse, and A. A.Menovsky,, Phys. Rev. B , 5375(R) (1987). M. B. Fontes, J. C. Trochez, B. Giordanengo, S. L. Budko,D. R. Sanchez, E. M. Baggio-Saitovitch, and M. A. Conti-nentino, Phys. Rev. B , 6781 (1999). T. H. K. Barron and G. K. White, Heat Capacity andThermal Expansion at Low Temperatures (Kluwer Aca-demic/Plenum,New York, 1999).28