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Specific Heat of Ce(1-x)La(x)RhIn(5) in Zero and Applied Magnetic Field A Very Rich Phase D

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Speci?c Heat of Ce 1?x La x RhIn 5in Zero and Applied Magnetic Field:A Very Rich Phase Diagram J.S.Kim,J.Alwood,D.Mixson,P.Watts,and G.R.Stewart Department of Physics,University of Florida,Gainesville,Fl.32611-8440Abstract:Speci?c heat and magnetization results as a function of ?eld on single-and poly-crystalline samples of Ce 1?x La x RhIn 5show 1.)a speci?c heat γof about 100mJ/moleK 2(in agreement with recent dHvA results of Alvers et al.);2.)upturns at low temperatures in C/T and χthat ?t a power law behavior (<=>Gri?ths phase non-Fermi liquid behavior);3.)a ?eld induced anomaly in C/T as well as M vs H behavior in good agreement with the recent Gri?ths phase theory of Castro Neto and Jones,where M?H at low ?eld,M?H λabove a crossover ?eld,C/T?T ?1+λat low ?eld,and C/T?(H 2+λ/2/T 3?λ/2)*exp(-μeff H/T)above the same crossover ?eld as determined in the magnetization and where λis independently determined from the temperature dependence of χat low temperatures,χ?T ?1+λand low ?elds.

I Introduction

Recently,a new family of heavy-fermion compounds has been discov-ered that crystallize in a layered,tetragonal structure with chemical composition CeMIn5,where M=Ir,Co,and Rh.Characteristic of heavy-fermion systems, each member exhibits a large Sommerfeld coe?cientγ(≡C/T as T→0)in the speci?c heat C.CeIrIn5and CeCoIn5are bulk superconductors1?2with transi-tion temperatures at T c=0.4K and2.3K and normal-state values ofγ≈750 mJ/molK2and1200mJ/molK2,respectively.CeRhIn5displays heavy-fermion antiferromagnetism with3T N=3.8K.A precise value ofγis di?cult to estab-lish unambiguously because of the N′e el order;a lower limit of approximately 400mJ/molK2has been quoted4?5.

In our high?eld speci?c heat measurements6on the CeMIn5com-pounds,we found that the large upturn for M=Rh in C/T above T N(C/T is already1000mJ/molK2at T N)as temperature is lowered appeared to be primarily due to magnetic interactions above the antiferromagnetic transition since the speci?c heat data at a given temperature for T>T N in di?erent?elds up to32T all coincide with one another when the temperature axis was scaled to T/T N.Recently Alver,et al.have performed7dHvA measurements on twelve single crystal samples spanning the whole composition range of Ce1?x La x RhIn5 and?nd rather low(i.e.inconsistent by approximately an order of magnitude with aγof400mJ/molK2)e?ective masses from the dilute Ce,large x end of the phase diagram up to x=0.1.At this Ce-rich end of the composition range they?nd an increase in the e?ective masses(which still remain≤10m e) which they ascribe to spin?uctuation e?ects.Alver,et al.conclude that the Ce f-electrons remain localized in Ce1?x La x RhIn5for all x,with the(modest) observed mass enhancement near pure CeRhIn5due to spin?uctuation e?ects. Although comparisons between speci?c heat and dHvA data have inherent prob-lems(not the least of which is the possibility of unseen,heavier mass orbits in the dHvA measurements),an e?ective mass enhancement of approximately ten normally corresponds to a speci?c heatγof only?50mJ/molK2.This is a wide discrepancy from the estimate of400mJ/molK2in4?5the literature;this discrepancy would be consistent with our high?eld speci?c heat result6that the upturn above T N in C/T in pure CeRhIn5is primarily caused by magnetic interactions,which would not cause a mass enhancement observable,e.g.,in dHvA measurements.

In order to help resolve this seeming disagreement,to determine the speci?c heatγ(also proportional to the e?ective mass)in a region of the phase diagram away from the antiferromagnetic anomaly,and to look for possible new behavior in the dilute limit we report here on a speci?c heat study of both single and polycrystalline samples of Ce1?x La x RhIn5,0≤x≤0.95.Certainly, doping studies8?10on other heavy Fermion systems,e.g.Ce1?x La x Cu2Si2, Ce1?x Th x Cu2Si2,and U1?x Th x Be13,have revealed interesting new information -both about the respective parent compound as well as new physics in the dilute limit.Polycrystalline samples were originally chosen for the study as being more easily and rapidly prepared.However,speci?c heat results for polycrystalline

Ce1?x La x RhIn5,x=0.5and0.8were determined to disagree with speci?c heat results for single crystal samples,while results agreed for x=0.15and0.95.This disagreement appears due to the presence of a second phase which we were able

to eliminate through long term annealing of the polycrystalline samples at a relatively low temperature.

II Experimental

Single crystal samples of Ce1?x La x RhIn5were prepared using the proce-dure described in ref.6,which was similar to that used in refs.4and7.Excess

In was removed from the resulting?at platelet crystals using an H2O:HF:H2O2 4:1:1etch which was di?erent than the centrifugal method(H2O:HCl4:1etch) used in ref.4(7);however the present work’s speci?c heat results(which are

a measure of bulk properties)should be relatively independent of such surface treatments.The polycrystalline samples in the present work(previous work

in the literature has been almost uniformly on single crystal samples)were prepared by melting together stoichiometric amounts of the appropriate high purity starting elements(using Ames Laboratory Ce and La,99.95%pure Rh from Johnson Mathey Aesar,and99.9999%In from Johnson Matthey Aesar

-the same starting materials as used for the single crystals)under a puri?ed inert Ar atmosphere.Weight losses after four melts,with a?ipping of the arc-melted button between melts to improve homogeneity,were in the range of1%, primarily due to In loss.Additional In was added in the beginning to correct for this,such that the In concentrations after the last melt were within±0.2%

of the stoichiometric amount.

Speci?c heat in?elds to13T were measured using established techniques11, while magnetic susceptibility data were measured in a SQUID magnetometer from Quantum Design.

III Results and Discussion

Figure1shows the speci?c heat divided by temperature vs temperature for single crystal Ce1?x La x RhIn5,x=0,0.15,0.5,0.8,and0.95and polycrystal Ce1?x La x RhIn5,x=0.32.All samples were single phase.Results for unan-nealed polycrystalline Ce1?x La x RhIn5,x=0.15and0.95,and annealed(35days

at720o C)polycrystalline Ce1?x La x RhIn5,x=0.5and0.8,were comparable to the single crystal results(see inset of Fig.1for an example);however,unan-nealed polycrystalline samples for x=0.5and0.8contained a second phase that ordered antiferromagnetically below1K.This was taken as a sign of an incipi-ent miscibility gap which-due to previous work being focussed on single crystal samples-was heretofore unknown.

From the data shown in Fig.1,one can follow the suppression of the an-tiferromagnetic transition with increasing La doping;there is a clear,although reduced in magnitude,transition at2K for15%La doping that is absent by

x=0.32.Although one might expect12non-Fermi liquid(’nFl’)behavior when

T N is suppressed to T=0,the temperature dependence of the C/T data for

C /T (m J C e -m o l e K

)T (K)

Figure 1:C/T vs T for Ce(1-x)La(x)RhIn(5)

x=0.32-although the

data show an upturn -is only measured for ?0.5K below the hump.This is

too restricted a temperature range to allow conclusions about the temperature dependence.

Before we discuss the behavior of γas a function of x in Ce 1?x La x RhIn 5,

we will ?rst focus on the upturn at low temperatures for x ≥0.5.

A Upturn in C/T for x ≥0.5

The upturn in C/T for x ≥0.5in Ce 1?x La x RhIn 5shown in Figs.1is

?t in Figs.2and 3for single crystalline,as well as single phase polycrystalline,material.Note in Fig.2that the data for the three di?erent samples agree rather well.There is certainly no sign in the dHvA results of Alver,et al.for a strong,heavy fermion upturn in C/T that would cause large e?ective masses.Thus,this upturn at low temperatures in C/T likely has a magnetic interaction explanation (see section C below for the ?eld dependence).The tempera-ture dependence of the upturns in C/T (see Figs.2and 3)for single crystal Ce 1?x La x RhIn 5,x=0.5,0.8,and 0.95,is not at all like the high temperature side of a Schottky peak (C ?1/T 2)but rather appears (in the somewhat limited temperature range that we have data)to follow C/T ?T ?1+λ,λC/T =0.63±0.1,0.37±0.1,and ?0respectively.This is the temperature dependence predicted for non-Fermi liquid behavior caused by disorder-induced spin clusters,the so-called Gri?ths phase 12?13.(Note that the ?ts of χto T ?1+λbelow 1.2K are much better than ?ts to either log T or T 0.5.)In this theory,the magnetic susceptibility

at low temperature should have the same power law dependence as C/T.

The susceptibility at low temperatures for these same compositions of single

C /T (m J C e -m o l e K )

T (K)

Figure 2:C/T for 3samples of x=0.5

C /T (m J C e -m o l e K )

T (K)

Figure 3:C/T vs T,x=0.5,0.8,0.95,?t to T?(-1+lambda)

(e m u /m o l e )T (K)

Figure 4:Chi vs T ?t to T?(-1+lambda)

crystal Ce 1?x La x RhIn 5,see Fig.4,does indeed ?t this T ?1+λtemperature dependence,with λχ={0.73,0.90},{0.50,0.70},{0.14,0.30}respectively for H {⊥, }the c-axis,where the absolute error bar for each value is ±0.1(with,however,somewhat better precision,useful for intercomparison between values derived from a given measurement technique.For example,0.14derived from χfor x=0.95is certainly less than 0.30derived for the other ?eld direction,but is comparable to the value of ?0derived for the same composition from the speci?c heat.)(Note that other standard non-Fermi liquid temperature dependences,such as χ?log T or T 0.5,do not ?t the χdata at all well.)Although for a given composition the respective exponents for C/T and χagree within experiment accuracy only for χ(H ⊥c),the recent theory 14of Castro Neto and Jones actually predicts that χand C/T may diverge di?erently at low temperature,relaxing the requirement of the early theory 12?13that λχ=λC/T .It is clear that the disorder requirement for uncompensated spins (which requires that M vs H is shows saturation behavior)is ful?lled for all these compositions (see discussion and accompanying ?gures in section C below.)In addition,the agreement in λC/T and λχfound for the upturn in C/T and χin the present work is comparable to that found by,e.g.,deAndrade et al.15in their study of Th 1?x U x Pd 2Al 3-even though they measured χdown to 0.5K,i. e.in a temperature range comparable to that for their speci?c heat measurements.The anisotropy of the susceptibility-determined λvalues is thought to be real,and not related to the discrepancy between λC/T and λχ.

As one possible check for a tendency towards magnetic behavior,the Wilson ratio (R ∝χ/γμ2eff )-which is used

16in the study of heavy Fermion systems to track the tendency towards magnetism,with R 0.8indicating 16magnetic behavior -for these Ce 1?x La x RhIn 5alloys is in the range of 1.0to 1.8,i.e.they de?nitely show magnetic character.As a further check for evidence

for spin clusters,we investigated these compositions for spin glass behavior and -to within the limits(±2%)of the accuracy of the measurements-found no di?erence between?eld cooled and zero?eld cooled data down to1.8K.This lack of observable spin glass behavior in the dc magnetic susceptibility in these samples does not rule out a Gri?ths phase interpretation17.

B Speci?c Heatγas a Function of x

The original goal of this work,besides the hope for new physics of inter-est in the dilute range(already partially ful?lled by the results discussed above for the low temperature upturn in C/T andχ)was to investigate the speci?c heatγ(de?ned as C/T as T→0)away from the region of the phase diagram where antiferromagnetism obscures C/T as T?→0in CeRhIn5diluted with La.As discussed above,after the antiferromagnetism is suppressed(x>0.15), a low temperature upturn in the C/T data(Fig.1)occurs that,normalized per Ce-mole,becomes more pronounced with increasing dilution of the Ce.This upturn appears not to be related to the e?ective masses measured by the dHvA measurements.

A further complication to determining the speci?c heatγis the rounded feature in C/T centered at?3K visible already for x=0.15above T N.As may be seen from Fig.5,the C/T data for x=0.5(triangles)and0.8(circles)in Ce1?x La x RhIn5above the low temperature upturn show a tendency to curve or bend downwards down to about1.5K,at which point the upturn discussed in the section above begins.This’hump’in C/T centered at?3K makes extrapo-lating C/T to T=0to determineγa somewhat imprecise procedure.It should be stressed that this rounded feature,or hump,in C/T has its provenance in the f-electron sublattice:such a feature is not present in C/T data for pure LaRhIn185.One possibility for correcting for this feature in order to determine γis to subtract o?both the low temperature upturn(see Fig.3for the?ts to the upturns)and a?t18to pure LaRhIn5and examine the remainder.As shown in the inset to Fig.5for x=0.5,this very rough approximation(the apparent negative value below about1K is,see Fig.3,merely a sign that the ?t to the upturn-which goes up to over1000mJ/Ce-moleK2at0.3K-is in error as T→1K)allows us to assign an approximate19γvalue per Ce mole of ≤100mJ/CemolK2for x≥0.5.This agrees much better with Alver,et al.’s dHvA results than the estimates of400mJ/CemolK2estimated4?5in the liter-ature.However,as the La dilution is removed,for x≤0.1,Alver,et al.report approximately a factor of two increase in e?ective mass due to spin?uctuation e?ects,with an e?ective mass for pure CeRhIn5that would correspond to aγof approximately50mJ/CemolK2.In the dilute limit,Alver et al.’s e?ective measured e?ective mass corresponds to aγof only25mJ/CemolK2.However, as may be seen in Fig.5,our C/T data at low temperature are much too ob-scured by the unexpected upturn as well as by the rounded maximum to supply any sort of accurate estimate forγbeyond the dilute,x≥0.5,range of≤100 mJ/CemolK2already quoted above.

C Field Induced Anomaly for x≥0.5

As a?nal aspect of new,unexpected behavior for CeRhIn5diluted with

C /T (m J C e -m o l e K )

T (K)

Figure 5:C/T vs T showing the ’hump’at 3K

La,when we were investigating the ?eld dependence of the upturn in the speci?c heat divided by temperature using magnetic ?eld as a probe,we discovered that applied ?eld suppresses the low temperature upturn in C/T at rather low ?eld and induces an peak in C/T that,with increasing ?eld,moves up in temperature and becomes broader and less pronounced.This rounded anomaly,shown in Fig.6for x=0.95(these data are typical of the results for all x ≥0.5)with ?eld in the basal plane (data in the perpendicular direction are within 15percent of these),is not that of either a spin glass (where C?1/T above the peak)or a Schottky anomaly (C?1/T 2above the peak)but rather seems to be a ?eld-induced anomaly.(The upturns in C/T for H ≥6T are caused by the applied ?eld splitting the nuclear magnetic moment energy levels and creating a Schottky peak in the speci?c heat.)

Castro Neto and Jones have recently published 14a theory of how the speci?c heat and magnetization of materials with non-Fermi liquid behavior caused by disorder-induced Gri?ths phase spin clusters should scale with mag-netic ?eld.In general,both the magnetization and speci?c heat are predicted to exhibit low ?eld behaviors (M ?H and C/T ?T ?1+λ)which crossover over to the respective high ?eld behaviors (M ?H λand C/T ?(H 2+λ/2/T 3?λ/2)e ?μeff H/T )at the same magnetic ?eld.The prediction for the ?eld and temperature de-pendence for the high ?eld speci?c heat leads to a peak in C/T (or a shoulder in

C)as a function of increasing temperature -thus qualitatively consistent with the data shown in Fig.6.

Although the speci?c heat data in ?eld was taken in fairly widely spaced ?elds,the fact that a peak occurs already in C/T in H=3T o?ers a prediction (the equality of the crossover ?eld requires that the crossover ?eld for the mag-netization data be perforce below 3T)that can be checked by examining the M vs H data,where a much more ?nely spaced sequence of ?elds was used.In

C /T (m J m o l e K )T (K)

Figure 6:Field-induced anomaly in C/T for x=0.95

addition,the high ?eld prediction that M ?H λcan be checked up to 5.5T,and this ?eld-dependence determination of λcan then be compared with that inde-pendently determined from the temperature dependence of χin Figure 4.Thus,magnetization data for both ?eld directions for single crystal Ce 0.05La 0.95RhIn 5are shown ?tted to these Gri?ths phase low and high ?eld predictions in Figures 7and 8,H ,⊥basal plane respectively.As may be seen,using the values for λχdetermined from Fig.4(0.14and 0.41for H( ,⊥)basal plane respectively)gives rather good 20agreement between the predicted,M ?H λdependence and the high ?eld magnetization data.(The ?t to the higher ?eld data with the lowest standard deviation actually gives λ=0.67;however,the standard devia-tions are within 8%of one another.)Further,the deviation from linear behavior at low ?elds occurs (see Figs.7and 8)above 0.8T and the deviation from the M ?H λpower law occurs below 1.2T.These estimates for the crossover ?eld are not inconsistent with the peak in C/T (where a peak is characteristic of the high ?eld regime)occuring in 3T,Fig.6.(Work under way 21to more thor-oughly characterize the low and high ?eld behavior for M and C/T for x=0.95has found that a peak in C/T ?eld data taken in 0.5T increments down to 0.3K ?rst appears at 1.5T.)

Another prediction 14of the Gri?ths phase theory of Castro Neto and

Jones,the ?eld and temperature dependence of C/T in the high ?eld limit,is compared 22to the 3T Ce 0.05La 0.95RhIn 5data (with the ?t 18to pure LaRhIn 5and the small,<10%at the lowest temperature,contribution due to the ?eld splitting of the nuclear moments,subtracted o?),H basal plane,in https://www.wendangku.net/doc/1c13469182.html,ing only two ?t parameters (the amplitude and the e?ective moment,μeff )and ?xing λ=0.14(based on λχ)gives the ?t (dashed line in Fig.9)as shown,with the reasonable 14,23?tted value for μeff (which corresponds to the average

M (e m u /m o l e )H (g auss)

Figure 7:M vs H for x=0.95,H in basal plane

M (e m u /m o l e )H (g auss)

Figure 8:M vs H for x=0.95,H perp.to basal plane

C /T (m J m o l e K )

T (K)

Figure 9:Fit of ?eld-induced anomaly to theory

moment in the Gri?ths phase spin cluster)of 1.25μB .Clearly,?tting C/T to (H 2+λ/2/T 3?λ/2)e ?μeff H/T is a fairly good representation of the data.(To give an idea how the ?t depends on the e?ective moment,a ?t to these 3T data with μeff constrained to be 1.0μB is shifted by to lower temperatures by ?0.2K from the present ?t.)

IV Conclusions

Despite the di?culty of precisely compensating for the broad peak in

C/T in Ce 1?x La x RhIn 5centered at about 3K,the apparent γper Ce mole for x ≥0.5,away from the antiferromagnetic transition in the phase diagram,appears to be less than 100mJ/Ce-moleK 2-in disagreement with estimates for γin the literature 4?5but not inconsistent with the dHvA results of Alvers et al.7There is a strong upturn in C/T below 1K for x ≥0.5that,when com-pared to the temperature dependence of the susceptibility and the non-linear M vs H data,is consistent with non-Fermi liquid behavior due to disordered spin clusters (’Gri?ths phases.’)Applied magnetic ?eld suppresses this upturn in C/T already by 3T;above 3T the C/T results show a broad anomaly that further broadens and moves to higher temperatures as ?eld is increased.This ?eld induced anomaly,together with the ?eld dependence of the magnetization,compares well with the predictions of the Gri?ths phase theory 14,24of Castro Neto and Jones,particularly in the magnetization data as a function of ?eld and the agreement of these data with the predicted λχexponent from the temper-ature dependence of the susceptibility.In summary,the breadth of behavior observed in Ce 1?x La x RhIn 5in zero and applied ?eld is indicative of a phase diagram of unusual richness and variety.

Acknowledgements:The authors wish to thank Antonio Castro Neto

for quite fruitful discussions.Work at the University of Florida by performed

under the auspices of the United States Department of Energy,contract no. DE-FG05-86ER45268.Partial summer support for J.Alwood and P.Watts from the NHMFL and University of Florida NSF REU programs respectively is gratefully acknowledged.

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17.For a discussion of spin cluster,spin glass,and Gri?ths phase behavior, see ref.12.

18.Mike Hundley,private communication.See also ref.4.

19.Note of course the unavoidable uncertainty is?tting the’hump’-which may very well involve entropy due to Ce-Ce interactions-to data from a more dilute composition and then applying this?t to more concentrated systems.

20.The”best?t”value for the exponentλfrom the?eld dependence of the magnetization for H basal plane shown in Fig.7is within0.1of the valueλχ=0.14determined from the temperature dependence ofχdetermined in Fig.4,i. e.within the error bar.For H⊥basal plane,the best?t to the magnetization data shown in Fig.8givesλ=0.67instead of the value determined from the temperature dependence ofχ,whereλχ=0.41.However, the standard deviation for the?t(to20data points)usingλχ=0.41is less than8%higher than that for the”best”?t.

21.J.S.Kim,J.Alwood,D.Mixson,and G.R.Stewart,to be published.

22.Fits to the6and9T data are similar,although the correction for the low temperature upturn in C/T caused by the nuclear hyper?ne level splitting due to the applied?eld is larger and the size of the?eld-induced anomaly in C/T with increasing?eld is rapidly decreasing.Since the crossover?eld between low and high?eld dependences,as determined by the magnetization,is?0.8-1.2 T,the3T data should be well in the high?eld limit.

23. A.H.Castro Neto,private communication.

24.Although a recent paper(https://www.wendangku.net/doc/1c13469182.html,lis,D.K.Morr,and J.Schmalian, Phys.Rev.Lett.87,167202{2001})has called the theory of Castro Neto and Jones into question based on dissipation arguments in the single impurity limit,an even more recent work by Castro Neto and Jones(cond-mat/0106176) argues that for concentrated systems the results of ref.14still hold.