Kondo effect in CeX c (X c =S, Se, Te) studied by electrical resistivity under high pressure
Y. Hayashi, S. Takai, T. Matsumura, H. Tanida, M. Sera, K. Matsubayashi, Y. Uwatoko, A. Ochiai
aa r X i v : . [ c ond - m a t . s t r- e l ] D ec Journal of the Physical Society of Japan
DRAFT
Kondo effect in CeX c (X c =S, Se, Te) studied by electrical resistivityunder high pressure Yuya Hayashi , Shun Takai , Takeshi Matsumura , , Hiroshi Tanida , Masafumi Sera , ,Kazuyuki Matsubayashi , Yoshiya Uwatoko , and Akira Ochiai Department of Quantum Matter, AdSM, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8530,Japan Institute for Advanced Materials Research, Hiroshima University, Higashi-Hiroshima, Hiroshima739-8530, Japan Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba, 277-8581, Japan Department of Physics, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan
We have measured the electrical resistivity of cerium monochalcogenices, CeS, CeSe, and CeTe, underhigh pressures up to 8 GPa. Pressure dependences of the antiferromagnetic ordering temperature T N , crystalfield splitting, and the ln T anomaly of the Kondo effect have been studied to cover the whole region fromthe magnetic ordering regime at low pressure to the Fermi liquid regime at high pressure. T N initiallyincreases with increasing pressure, and starts to decrease at high pressure as expected from the Doniach’sdiagram. Simultaneously, the ln T behavior in the resistivity is enhanced, indicating the enhancement of theKondo effect by pressure. It is also characteristic in CeX c that the crystal field splitting rapidly decreasesat a common rate of − . f state and furtherenhancement of the Kondo effect. It is shown that the pressure dependent degeneracy of the f state is akey factor to understand the pressure dependence of T N , Kondo effect, magnetoresistance, and the peakstructure in the temperature dependence of resistivity.
1. Introduction
Through the hybridization between conduction elec-trons and localized f state ( c - f hybridization), the f electrons at different atomic sites interact with eachother, leading to a magnetic ordered state. This isthe Ruderman-Kittel-Kasuya-Yosida (RKKY) interac-tion. At the same time, the c - f hybridization causes thelocal spin and orbital degrees of freedom to be screenedby the conduction electrons, leading to the reduction ofthe local moment and the magnetic ordering tempera-ture. This is the Kondo effect. The strength of the c - f hybridization, J cf , is the primary parameter to describethese competing phenomena. In Ce based compoundswith one 4 f electron, the RKKY interaction dominatesthe Kondo effect when J cf is small, and the magnetic or-dering temperature increases with increasing J cf . Withfurther increasing J cf , the Kondo effect gradually dom-inates the RKKY interaction, and the magnetic order-ing temperature starts to decrease. Finally, the magneticordering vanishes, and the c - f hybridized system formsa Fermi liquid ground state with heavy effective mass.This general scheme has been described by Doniach’sdiagram.
1, 2
Of particular interest in recent years is inthe vicinity of J cf where the magnetic ordering vanishes,which is called the quantum critical point (QCP). In thisregion, the order parameter fluctuation causes the physi-cal properties to deviate from the Fermi liquid behavior, and in some cases causes unconventional superconductiv-ity. In the course of this study, application of pressureis an suitable method to experimentally realize the vari-ation of J cf .CeS, CeSe, and CeTe, crystallizing in a cubic NaCl-type structure, exhibit antiferromagnetic (AFM) order-ings with q = ( , , ) at T N =8.4, 5.4, and 1.9 K, respec-tively. In a cubic crystalline electric field (CEF), theHund’s rule ground multiplet of J = 5 / doublet and the Γ quartet. It is already well estab-lished that the ground state of CeX c is the Γ and theenergy level of the Γ state is at 140 K, 116 K, and 32K for CeS, CeSe, and CeTe, respectively. The mag-nitude of the ordered moment in the AFM phase is 0.57 µ B , 0.56 µ B , and 0.3 µ B for CeS, CeSe, and CeTe, re-spectively. The reduction of the ordered moment fromthe expected value of 0.71 µ B for the Γ ground state canbe ascribed to the Kondo effect. However, the largest re-duction in CeTe cannot be understood from the Kondoeffect only, because the Kondo effect is the weakest inCeTe and the strongest in CeS as judged from the ln T anomaly ( dρ/d ln T ) in resistivity. This is still an openquestion.In most of the Ce based compounds that have beenstudied to date under high pressure in the quantumcritical region, the CEF ground states are well isolatedKramers doublets, where only the magnetic dipolar de-
1. Phys. Soc. Jpn.
DRAFT gree of freedom plays a role. It is also the case in CeX c at ambient pressure with the Γ ground state. However,it has been strongly suggested by recent experiments onCeTe that the Γ excited level falls down under highpressure.
12, 13
The CEF splitting is expected to vanish ataround 2.5 GPa in CeTe. This means that the quadrupo-lar degree of freedom of the Γ state also takes part in thephysical properties at low temperatures. It has alreadybeen shown for CeTe that an antiferroquadrupolar inter-action through the Γ excited level is essential to under-stand the properties under high pressure. Therefore,nontrivial phenomenon could take place in the quantumcritical region by an interplay between nondipolar de-grees of freedom and the Kondo effect. This is also animportant issue in relation to the recent studies in Prbased compounds, where the role of quadrupolar fluctu-ation for the superconductivity attracts interest. Highpressure study on CeX c , therefore, has its significance instudying the possibility of multipolar degrees of freedomto participate in the quantum critical phenomenon.Another advantage in studying CeX c lies in its sim-ple electronic structure. The localized f state hybridizeswith the p -orbital of X c , which forms the valence band.The conduction band is formed by the 5 d -orbital of Ce,containing one conduction electron per formula unit. The f level is located ∼ . c have been clarified in detail bythe angle-resolved photoemission spectroscopy and thede Haas-van Alphen effect measurements. In the present work, we have measured the electricalresistivity of CeX c under high pressures up to 8 GPato study the enhancement of the Kondo effect and thecrossover from the magnetic ordering regime to the Fermiliquid regime. We show that the ln T behavior in the re-sistivity is enhanced by applying pressure in all the CeX c compounds. The pressure dependences of T N , CEF split-ting, coefficient of the ln T term, and the temperatures ofresistivity maximum below which the Fermi liquid stateis formed, are extracted from the data. Although theoverall feature can be understood in the framework ofDoniach’s diagram, we suggest that the level loweringof the Γ excited state at high pressures gives rise tosome distinct phenomena in CeX c in the crossover re-gion around the QCP. We compare these results withthose of other inter-metallic compounds.
2. Experiment
The single crystalline samples were prepared by Bridg-man method as described in Ref. 16. Resistivity un-der high pressure was measured by normal four ter-minal method. The current direction was along the[001] axis. For measurements up to 2.5 GPa, we useda CuBe/NiCrAl hybrid-piston-cylinder cell with daphneoil as a pressure transmitting medium. Magnetic fieldwas also applied using a 15 T cryomagnet. The field di- rection was along the [001] axis, which was parallel tothe current direction. The measurements up to 8 GPawere performed at zero field using a cubic anvil cell withfluorinert as a pressure transmitting medium.
3. Experimental Results
Figure 1 (a) shows the temperature ( T ) dependences ofthe resistivity ( ρ ) of CeTe at several pressures. The ρ ( T )of LaTe is also shown for reference to estimate the con-tribution from phonon scattering. At ambient pressure,a hump anomaly is observed around 20 K, reflecting theCEF splitting; ρ decreases below ∼
20 K because themagnetic scattering through the Γ exited state grad-ually vanishes. We name this characteristic temperature T ∗ . With further decreasing temperature, ρ drops steeplyat 2 K, reflecting the AFM order. The ln T behavior in ρ ( T ) due to the Kondo effect can be recognized above T ∗ after subtracting the ρ ( T ) of LaTe, whereas it is not rec-ognizable below T ∗ . These behaviors are consistent withthe previous study. With increasing P , ρ ( T ) exhibits a clear ln T depen-dence below ∼
100 K even before subtracting the ρ ( T )of LaTe. This shows that the Kondo effect (single siteKondo scattering) is enhanced by the pressure. If we sub-tract the ρ ( T ) of LaTe, the ln T dependence continuesup to 300 K. The hump anomaly at T ∗ shifts to lowertemperatures, corresponding to the level lowering of theΓ exited state. However, above 2.5 GPa, the humpanomaly is no more distinguishable from the peak emerg-ing around 6 K, probably reflecting the Kondo coherence.We name this temperature of resistivity maximum as T m1 hereafter. When P is increased from 3.5 GPa to 5 GPa,the coherence peak slightly shifts to higher temperature,which seems to reflect the increase in the Kondo tem-perature T K . However, the peak strangely shifts againto lower temperature at 7.3 GPa, which is opposite tothe normal P dependence of the coherence peak in Cebased compounds. Although this could be somehow re-lated with the transition to the CsCl structure at 8 GPain CeTe, the reason is unclear.The low temperature part of ρ ( T ) around T N is shownin the inset of Fig. 2 (a). As indicated by the arrows, T N of CeTe initially increases with increasing P , andthen starts to decrease above ∼ ρ ( T ) corresponding to T N is no more able tobe identified. The P dependence of T N is plotted in Fig. 2(a). T N determined from the specific heat measurementin Ref. 13 is also plotted. With respect to the successivetransitions at 1.2 GPa and 1.8 GPa reported in Ref. 13,the higher transition temperature agree with T N deducedfrom ρ ( T ). Anomaly corresponding to the second tran-sition at a lower temperature, which is indicated by thefilled triangles in Fig. 2 (a), was not detected in ρ ( T ) as
2. Phys. Soc. Jpn.
DRAFT r ( m W ¥ c m ) T (K) CeTe
LaTe0 GPa122.53.5 56.57.3 (a) T * T m1 T N r ( m W ¥ c m ) T (K) CeSe
LaSe012345 6 7 GPa 8 (b) T m2 T m1 T * T N r ( m W ¥ c m ) T (K) CeS
LaS01.5 22.534 4.55 GPa6.5 8 (c) T m1 T m2 T * T N Fig. 1. (Color online) Temperature dependences of the electrical resistivity of (a) CeTe, (b) CeSe, and (c) CeS, under high pressures.The resistivity of LaTe, LaSe, and LaS, at ambient pressure, are also shown for reference. T N ( K ) r ( m W ¥ c m ) (+60)(+50)(+40)(+30)(+15) (+5) Fig. 2. (Color online) Pressure dependences of T N of (a) CeTe, (b) CeSe, and (c) CeS, determined from ρ ( T ) (open circles). The insetof (a) shows ρ ( T ) of CeTe in the vicinity of T N . Open and filled triangles in (a) represent the transition temperatures of CeTe determinedfrom the specific heat measurement in Ref. 13. The filled triangles at 1.2 GPa and 1.8 GPa show the successive transitions. Open circlesin (b) correspond to the weak anomalies observed in ρ ( T ) of CeSe below T N . a clear anomaly. The critical pressure, P c , is estimatedto be from 3 to 3.5 GPa, although this requires a furtherstudy at lower temperatures. Figure 1 (b) shows the ρ ( T ) curves of CeSe up to 8GPa. The ρ ( T ) of LaSe is also shown for reference to es-timate the contribution from phonon scattering. At am-bient pressure, in the same way as in CeTe, the humpanomaly reflecting the CEF splitting appears at around T ∗ = 60 K and the kink anomaly due to the AFM orderis clearly observed at 5.4 K. The ρ ( T ) weakly increases with decreasing T below ∼
20 K down to T N . This in-crease can be ascribed to the Kondo effect within theΓ ground state. The ln T behavior in the high- T regionabove T ∗ can be recognized after subtracting the ρ ( T ) ofLaSe. These behaviors at ambient pressure are consistentwith the previous study. With increasing P , ρ ( T ) increases and exhibits a clearln T anomaly below ∼
200 K, which shows that theKondo effect (single site Kondo scattering) is enhancedby the pressure. If we subtract the ρ ( T ) of LaSe, theln T dependence continues up to 300 K. Simultaneously,the hump anomaly around T ∗ shifts to lower tempera-
3. Phys. Soc. Jpn.
DRAFT tures, indicating that the energy level of the Γ exitedstate falls down with increasing P . Above 5 GPa, T ∗ is no more recognizable because it is hidden behind thesignificant increase of ρ ( T ) due to the Kondo effect.The ln T slope below ∼
20 K down to T N , where theKondo effect within the Γ ground state dominates, alsoincreases with P . Above 5 GPa, it is more enhanced and ρ ( T ) exhibits a steep increase down to T N , where a sharpcusp is observed. These behaviors are in contrast to CeTeat high pressures, where ρ ( T ) follows a single ln T slopedown to T m1 ∼ T N soonbecomes unclear at high pressures above 2 GPa.In CeSe, as indicated by the arrows in Fig. 1 (b), T N initially increases with P , taking a maximum at ∼ P dependence of T N is plottedin Fig. 2 (b). The critical pressure, P c , is estimated to bearound 7.5 GPa. Above 3 GPa, we can also see anotherweak anomaly in ρ ( T ) below the main cusp anomaly at T N , which is plotted by the open circles. We speculatethat this transition reflects some change in the mag-netic structure such as the one we discussed in CeTeby a mean-field calculation. Namely, in the intermedi-ate phase, the AFM moment is possibly oriented alongthe [100] axis because of the Γ level lowering, and inthe low- T phase, the direction changes to the [111] axis,reflecting the easy axis of the Γ ground state.All these behaviors change significantly above 7 GPa,where ρ ( T ) at low temperatures decreases significantly.At 7 GPa, a new broad peak appears at around 6 K,which we suggest the peak of T m1 , the same kind ofKondo coherence peak as the one observed in CeTe. Theupturn in ρ ( T ) below 3 K probably suggests the AFM or-dering existing below the lowest temperature of 2.5 K. Itis also remarked that there is a broad hump anomaly ataround 60 K, which we name T m2 . At 8 GPa, the positionof this peak shifts to around 70 K, above which the ln T behavior is still observed. At low temperatures below 70K, ρ ( T ) shows a more significant decrease down to thelowest temperature, suggesting the formation of a Fermiliquid state. Therefore, the broad peak at T m2 , 60 K at 7GPa and 70 K at 8 GPa, may also be assigned to anotherKondo coherence peak. The coherence peak at T m1 stillseems to exist at around 15 K, where a broad hump isobserved. Thus, the double peak structure in ρ ( T ) is thecharacteristic of CeSe in the Fermi liquid regime, andboth T m1 and T m2 shift to higher temperatures with in-creasing P . Figure 1 (c) shows the ρ ( T ) curves of CeS at highpressures. At ambient pressure, in the same way as inCeTe and CeSe, the hump anomaly reflecting the CEFsplitting appears at around T ∗ = 70 K and the cuspanomaly due to the AFM order is clearly observed at8.7 K. An increase in ρ ( T ) with decreasing T is observed below ∼
20 K down to T N , which can be ascribed to theKondo effect within the Γ ground state. The slope ofthe ln T dependence in the low- T region is larger thanthat of CeSe. The ln T dependence is also visible in thehigh- T region above T ∗ even before subtracting the ρ ( T )of LaS. The largest slopes of the ln T dependences inCeS at ambient pressure, both in the high- and low- T regions, show that the c - f hybridization is the strongestamong the three compounds of CeX c . These behaviors atambient pressure are consistent with the previous study. With increasing P , the ln T slope increases as in CeTeand CeSe, reflecting the enhancement of the Kondo effect(single site Kondo scattering). A drastic phenomenonin CeS is that ρ ( T ) in the low- T region increases moresteeply than the ln T slope in the high- T region. Thisbehavior becomes more significant above 2.5 GPa. Theincrease in ρ ( T ) on approaching T N , which can probablybe ascribed to the enhancement of the Kondo effect, ismuch larger than that of CeSe. With respect to T N , itinitially increases with increasing P , followed by a max-imum around 2 GPa, and decreases to 4.5 K at 4 GPa.The upturn in ρ ( T ) of 3 GPa below 3.5 K probably sug-gests another AFM ordering existing below the lowesttemperature of 2.8 K. The P dependence of T N is plot-ted in Fig. 2 (c). The critical pressure, P c , is estimatedto be around 4.5 GPa.Above 4.5 GPa, a significant change takes place atlow temperatures. Although the ln T behavior remainsat high temperatures above 100 K, ρ ( T ) at low temper-atures decreases significantly with increasing P . At 4.5GPa, a new broad peak appears at around 12 K, whichwe suggest the peak of T m1 , the same kind of Kondo co-herence peak as the one observed in CeTe and in CeSe.It is again remarked that there is another broad humpanomaly at around 100 K, which we name T m2 . Withfurther increasing P , both of these broad peaks shift tohigher temperatures. Finally at 8 GPa, the ρ ( T ) curvebecomes like that of a typical Fermi liquid system. How-ever, the two broad peaks at T m1 and T m2 still remain in ρ ( T ) at 8 GPa; one is at around 60 K and the other isat around 200 K, both of which may be ascribed to thecoherence peak in the same way as in CeSe. Figure 3 shows the temperature dependences of electri-cal resistivity in magnetic fields and under high pressuresfor CeTe, CeSe, and CeS. All the compounds generallyexhibit negative magnetoresistance (MR). In CeTe, theMR effect increases significantly with increasing P . At1.5 GPa and 2.5 GPa, where the Kondo effect is clearlyrecognized in the ln T behavior in ρ ( T ) at zero field, the ρ value decreases and the peak in ρ ( T ) at T m1 , 7 K and5 K, respectively, shifts to higher temperatures with in-creasing the field. This is a characteristic behavior ob-served in concentrated Kondo systems.
19, 20
In CeSe and
4. Phys. Soc. Jpn.
DRAFT T (K) T (K) r ( m W ¥ c m ) T (K)
CeS r ( m W ¥ c m ) CeSe r ( m W ¥ c m ) CeTe
06 10 14.5 T
Fig. 3. (Color online) Temperature dependences of electrical resistivity in magnetic fields and under high pressures for CeTe, CeSe,and CeS. Arrows indicate the temperatures of antiferromagnetic ordering. T (K)CeS1 T (K)CeSe1.00.50 — ( r . — r ) / r T (K)CeTe
Fig. 4. (Color online) Temperature dependences of the magnetoresistance at 14.5 T, − ( ρ . − ρ ) /ρ , at several pressures. CeS, by contrast, although the MR effect increases withincreasing P , it is not so significant in comparison withthat of CeTe. This is because the system is still wellinside the magnetic ordering regime at these pressuresbelow 2.5 GPa. The Kondo temperature T K for the Γ ground state may be much smaller than T N . In CeTe, onthe other hand, T K is expected to be already larger than T N at pressures of 1.5 GPa and 2.5 GPa because of thehigh degeneracy of the ground state due to the almostcollapsed CEF.
21, 22
To describe the MR associated with the Kondo effectin more detail, we show in Fig. 4 the temperature depen-dences of the relative MR at 14.5 T for several pressures.We focus on the temperature range well above T N : T >
5. Phys. Soc. Jpn.
DRAFT
K for CeTe and
T >
10 K for CeSe and CeS. The magni-tudes of MR in CeSe and CeS are almost the same. If weobserve in detail, however, MR is almost independent of P below 2.5 GPa in CeSe, whereas in CeS MR increasesslightly with increasing P .It is noted that CeTe has the largest MR among thethree compounds in spite of the fact that J cf is the small-est. This may be because the f -state degeneracy influ-enced by the low-lying Γ excited level, which increases T K , is the largest among the three compounds even atambient pressure. With increasing P , the MR in CeTeincreases below ∼
15 K, and it becomes almost indepen-dent of P above 1.5 GPa. It is also remarked that theMR does not change with P above ∼
15 K.Finally, we comment on the magnetic ordering. InCeTe, T N exhibit a significant dependence both on theapplied field and pressure, which can be explained byconsidering the magnetic and quadrupolar interaction inaddition to the Γ level lowering as we have shown inthe previous study.
12, 13
In CeSe and CeS, by contrast, T N is little affected by the field in this P range below 2.5GPa. This is because the energy scale of the magnetic in-teraction is larger than that of CeTe, resulting in larger T N , and also because the Γ level is still located at highenergy and therefore the level mixing does not affect themagnetic ordering. Finally, although this is not a subjectof this work, weak anomalies observed in CeS below T N in Fig. 3, which are clearly identified at 0 GPa and 1.5GPa in magnetic fields, are those reflecting the changein the magnetic structure or the domain distribution. We can see this phase boundary also in CeSe at 3.5 Kin the data at 0 GPa and 10 T. This anomaly moves to
H <
10 T at 1.5 GPa and 2.5 GPa, which are not shownin Fig. 3.
4. Discussion
In Ce compounds, the coefficient of the ln T term inthe magnetic part of the resistivity, dρ mag /d ln T ≡ S ,is generally associated with J cf D ( ε F ), where D ( ε F ) rep-resents the density of states of the conduction electronsat the Fermi energy. When the system is in the mag-netic ordering regime in the Doniach’s diagram, as in thepresent case of CeX c , the coefficient S and the ρ valueat low temperatures generally increase with increasing P . In addition, in this regime, a double peak structureis observed in ρ mag ( T ) in many systems. The examplesare CeAl , CeCu , CeAu Si , CeNiGe , Ce Pd Si , andCe Ni Ge .
5, 23–28
The peak at high temperature is gen-erally interpreted as reflecting the CEF level splitting ofthe f state and the second peak at low temperature asthe formation of a coherent Kondo state. When the sys-tem is near the boundary between the magnetic orderingregime and the Fermi liquid regime, the ρ mag ( T ) gener-ally exhibit a single peak at around the Kondo coherencetemperature, and the peak shifts to high temperatures with increasing P without changing the peak height. Thepeak position can finally reach to 300 K. The peak ofCEF seems to be merged with the coherence peak. Theexamples are CeIn , CeAl and CeCu .
4, 29–31
To discussthe pressure dependence of these characteristic temper-atures of CeX c , in Fig. 5 we plot T N , T ∗ , T m1 , T m2 , andthe slope S , which have been deduced from ρ mag ( T ), as afunction of pressure. ρ mag ( T ) was obtained by subtract-ing ρ ( T ) of LaX c . We discuss these parameters in thefollowing. The most important and distinctive featurewe consider in CeX c is the shrinking of the CEF underpressure, which is reflected in the P dependence of T ∗ .This effect leads to the increase in the f -state degeneracyand, therefore, the increase in T K .
21, 22
First, we discuss the P dependence of the ln T termin ρ ( T ). When the CEF anomaly around T ∗ is at highenough temperatures, we can extract two different coef-ficients of the ln T term from high- T and low- T regionsabove and below T ∗ , which we represent as S H and S L ,respectively. Pressure dependences of these coefficientsare also shown in the bottom panels of Fig 5. In CeTe,since T ∗ is soon merged into a peak at T m1 , only S H isshown. In CeSe and CeS, the two regions are clearly sep-arated up to the critical pressure P c where T N vanishes.With respect to the pressure dependence of S H , they al-most coincide with each other at high pressures, although S H of CeS is slightly larger than others at P = 0. We alsosee that the increasing rate of S H with pressure is almostthe same in CeX c . On the other hand, S L for CeS isalmost ten times larger than that of CeSe and S L forCeTe is unable to recognize, which is probably becausethe Kondo scattering in CeTe is too small and is hiddenbehind the magnetic scattering due to disordered mo-ments. This difference in S L shows that the Kondo effectin the low- T region within the Γ ground state is thestrongest in CeS and the weakest in CeTe. By contrast,the similar values of S H seems to suggest that the Kondoeffect in the high- T region is more influenced by the f -state degeneracy involving all the CEF states than the c - f hybridization effect.Another point to be noted is that S L exceeds S H inCeSe and in CeS at high pressures. Theoretically, theln T coefficient is associated with the degeneracy, λ , ofthe f state and is proportional to λ − In the high- T region, λ = 6, and in the low- T region when only theΓ ground state is thermally populated, λ = 2. There-fore, S H is generally larger than S L , which is actuallythe case in many other inter-metallic compounds.
5, 23
Inthis sense, CeSe at low pressures is normal. Although thelarger increasing rate of dS L /dP than dS H /dP is also ob-served in CeAl , it is particularly anomalous in CeSeand CeS that S L exceeds S H .Next, with respect to T N , T ∗ , T m1 , and T m2 , a generalbehavior observed in other inter-metallic compounds inthe magnetic ordering regime can be summarized as the
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DRAFT T N , T * , T m , T m ( K ) (a) CeTeT N T m1 T * 806040200 (b) CeSeT * T m2 T N T m1 m2 T N T * T m1 d r / d l n T Fig. 5. (Color online) (Top panels) Pressure dependences of the characteristic temperatures T N , T ∗ , T m1 , and T m2 . The lines areguides for the eye. It is remarked that all the three lines for T ∗ are expressed as T ∗ ( P ) = T ∗ (0) − . P . (Bottom panels) Pressuredependences of the coefficient of the ln T term in the magnetic part of the resistivity, dρ mag /d ln T , which are estimated in the high- T region above T ∗ or T m2 and in the low- T region between T N and T ∗ or between T m1 and T m2 . following (see Fig. 3 of Ref. 5 or Fig. 2 of Ref. 23):(1) Double peaks appear at T ∗ and T m1 ( T ∗ > T m1 ).With increasing P , T ∗ slightly decreases and T m1 slightly increases.(2) Around P c , T m1 starts to increase rapidly andmarge with T ∗ . Then, a single peak remains in ρ ( T )and T ∗ is renamed to T m2 , below which the Fermiliquid state is formed.(3) T m2 increases with increasing P .These behaviors are actually reproduced by a theory in-cluding the CEF effect. However, the behavior of thesecharacteristic temperatures in CeX c is slightly different:(1) T ∗ decreases with increasing P much more rapidlyat a rate of − . T ∗ and T m2 seems not to be connected in CeSe andin CeS. Double peak structure is still observed above P c .The slight decrease of T ∗ with P in general cases meansthat the true CEF level does not actually change with P .In the present cases of CeX c , however, the rapid decreaseof T ∗ with P seems to show that the actual CEF split-ting decreases with P .
12, 13
This is the most importantand distinctive feature we consider in CeX c , which is as-sociated with the f -state degeneracy and the increase in T K . This effect might also be associated with the anoma-lous increase of S L at high pressures.The critical pressure P c could also be affected by thepressure dependent CEF splitting. From Fig. 2 we seethat P c is the largest in CeSe and the lowest in CeTe.Although the P dependence of T N of CeX c each followsthat of the Doniach’s diagram, this sequence of P c doesnot coincide with the that of the strength of J cf , whichis the smallest in CeTe and the largest in CeS. Here, itis necessary to take into account the difference in theCEF splitting, i.e., the largest in CeS and the smallest inCeTe. Especially in CeTe, the CEF splitting soon van-ishes at around 2 GPa, where T N starts to decrease. Thesmallest P c in CeTe can be understood by consideringthe increased degeneracy of the f state due to the col-lapsed CEF, which results in the increase of T K thanthat of CeSe and CeS. Then, T N of CeTe vanishes at thesmallest pressure among the three compounds.We consider that the pressure dependence of T N inFig. 2 can be understood as a result of three effects.Firstly, with increasing P , J cf increases, which results inthe increase in the RKKY interaction and T N . Secondly,the Γ CEF level falls down, which also contributes tothe increase in T N because the Γ state can give rise tolarger magnetic moment. This is explained in Ref. 13 forCeTe by using a mean-field model calculation. These twoeffects both contribute to the increase in T N . Thirdly, T K increases with P through the increase of both J cf and the f -state degeneracy. In the present case of CeX c , we sug-
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DRAFT gest that the latter effect influences T K more than J cf .The MR effect can also be roughly explained. In CeSeand CeS below 2.5 GPa, the Γ state is still well sepa-rated and the MR effect reflects the suppression of theKondo effect within the Γ ground state by the appliedfield. To understand the weak MR effects in CeSe andCeS shown in Fig. 4, Fig. 2 of Ref. 20 based on a peri-odic Anderson model is a good reference. With increasing P , T K of the Γ ground state increases, and the scaledparameters of T /T K and H/T K decrease. Since we arein the region of T /T K >
1, the increase of MR by thedecrease of
T /T K would be cancelled by the decrease of H/T K . Then, the MR effect is expected to be weak inCeSe and CeS. To analyze in more detail to understandthe larger P dependence of MR in CeS than in CeSe, itis necessary to consider that J cf is larger in CeS than inCeSe (see S L in Fig. 5) and also that the pressure regionof 0 ∼ . P c , or the pressurewhere S L takes the maximum. It is also necessary to takeinto account the gradual increase of f state degeneracyat high pressures.The much larger MR effect in CeTe should be morerelated to the higher degeneracy due to the small CEFsplitting, which almost vanishes above 1.5 GPa. Al-though J cf in CeTe is the smallest among the three com-pounds, the larger degeneracy of the f -state enhances theKondo effect and leads to the large MR effect. It is re-marked that, in Fig. 4, MR of CeTe increases with P upto 1.5 GPa and becomes almost independent of P above1.5 GPa. This coincides with the shrinking process ofCEF which is demonstrated by the variation of T ∗ inFig. 5(a). Above 1.5 GPa, the degeneracy of the groundstate do not increase any more with P , and then the MReffect changes little with P , which may be understood inthe same way as in CeSe and in CeS by the decrease of T /T K and H/T K . Above 15 K, where the Γ excited levelis well populated, the f state can be recognized as beingfully degenerate, and the MR shows little P dependence.Finally, although we have studied a wide region of pres-sure, temperature, and magnetic field space, we remarkthat there still remain a low temperature region below2 K that should be studied in future. As shown by the ρ ( T ) curves in Fig. 1, the ρ values at 2 K are still muchhigher than the residual resistivities, which are expectedfrom the extrapolation of the ρ ( T ) curves at 0 GPa. Thisshows that the system has not yet fallen to the groundstate, especially near pressures around P c , and there re-main a possibility that some drastic phenomena occurbelow 2 K.
5. Conclusion
We have studied the Kondo effect in CeS, CeSe, andCeTe by the electrical resistivity measurements underhigh pressures up to 8 GPa. The ln T term in the temper-ature dependence of the resistivity commonly increases with increasing pressure, indicating that the c - f hy-bridization increases with pressure. There are two ln T regions. One is at high temperatures, where both Γ andΓ states are involved, and the other is at low temper-atures, where the Γ ground state is mainly involved,resulting in the peak structure in ρ ( T ) around T ∗ re-flecting the CEF splitting. From the ln T term in the lowtemperature region, we see that J cf is the largest in CeSand the smallest in CeTe.CeX c under high pressure starts from the magnetic or-dering regime and the pressure dependence of T N can ba-sically be understood from the Doniach’s diagram. How-ever, it is necessary to take into account the significantdecrease and collapse of the CEF splitting under pres-sure to interpret the sequence of the critical pressure,where T N vanishes; P c (CeTe) < P c (CeS) < P c (CeSe),which contradicts with the sequence of J cf . We suggestthat the increase in T K under high pressure, which isbrought about more by the increase of the f -state de-generacy than by the increase of J cf , is responsible forthe pressure dependence of T N and magnetoresistance.Finally, the appearance of the double peak structure in ρ ( T ) in CeSe and CeS at high pressures near and above P c is also a distinctive feature in CeX c . Since T ∗ seemsto be connected to the low-temperature peak at T m1 ,the second peak at T m2 may suggest an appearance ofanother energy scale associated with the formation ofthe Kondo singlet state. In any case, however, this is anopen question to be studied in future. Acknowledgements
This work was carried out by the joint research in theInstitute for Solid State Physics, the University of Tokyo,and was supported by JSPS KAKENHI Grant Numbers2430087 and 15K05175.
1) S. Doniach, Physica B+C , 231 (1977).2) J. Otsuki, H. Kusunose, and Y. Kuramoto, J. Phys. Soc. Jpn. , 034719 (2009).3) H. v. L¨ohneysen, A. Rosch, M. Vojta, and P. W¨olfle, Rev. Mod.Phys. , 1015 (2007).4) G. Knebel, D. Braithwaite, P. C. Canfield, G. Lapertot, andJ. Flouquet, Phys. Rev. B , 024425 (2001).5) Z. Ren, L. V. Pourovskii, G. Giriat, G. Lapertot, A. Georges,and D. Jaccard, Phys. Rev. X , 031055 (2014).6) F. Hulliger, B. Natterer, and H. R. Ott, J. Magn. Magn. Mater. , 87 (1978).7) H. R. Ott, J. K. Kjems, and F. Hulliger, Phys. Rev. Lett. ,1378 (1979).8) J. Schoenes and F. Hulliger, J. Magn. Magn. Mater. ,43 (1987).9) J. Rossat-Mignod, J. M. Effantin, P. Burlet, T. Chattopadhyay,L. P. Regnault, H. Bartholin, C. Vettier, O. Vogt, D. Ravot,and J. C. Achart, J. Magn. Magn. Mater. , 111 (1985).10) A. D¨onni, A. Furrer, P. Fisher, S. M. Hayden, F. Hulliger, andT. Suzuki, J. Phys: Condens. Matter , 1119 (1993).11) A. D¨onni, A. Furrer, P. Fisher, and F. Hulliger, Physica B8. Phys. Soc. Jpn. DRAFT186-188 , 541 (1993).12) Y. Kawarasaki, T. Matsumura, M. Sera, and A. Ochiai, J. Phys.Soc. Jpn. , 023713 (2011).13) H. Takaguchi, Y. Hayashi, T. Matsumura, K. Umeo, M. Sera,and A. Ochiai, J. Phys. Soc. Jpn. , 044708 (2015).14) K. Matsubayashi, T. Tanaka, A. Sakai, S. Nakatsuji, Y. Kubo,and Y. Uwatoko, Phys. Rev. Lett. , 187004 (2012).15) M. Nakayama, H. Aoki, A. Ochiai, T. Ito, H. Kumigashira, T.Takahashi, and H. Harima, Phys. Rev. B , 155116 (2004).16) M. Nakayama, N. Kimura, H. Aoki, A. Ochiai, C. Terakura,T. Terashima, and S. Uji, Phys. Rev. B , 054421 (2004).17) G. Chiaia, L. Du`o, O. Tjernberg, M. G¨othelid, M. Bj¨orkqvist,H. Kumigashira, S.-H. Yang, T. Takahashi, T. Suzuki, and I.Lindau, Phys. Rev. B , 12030 (1998).18) J. M. Leger, R. Epain, J. Loriers, D. Ravot, and J. Rossat-Mignod, Phys. Rev. B , 7125 (1983).19) A. Takase, K. Kojima, T. Komatsubara, and T. Kasuya, SolidState Commun. , 461 (1980).20) N. Kawakami and A. Okiji, J. Phys. Soc. Jpn. , 2114 (1986).21) K. Yamada, K. Yosida, and K. Hanzawa, Prog. Theor. Phys. , 450 (1984).22) K. Hanzawa, K. Yamada, and K. Yosida, J. Magn. Magn. Mater. , 357 (1985).23) H. Miyagawa, G. Oomi, M. Ohashi, I. Satoh, T. Komatsubara,M. Hedo, and Y. Uwatoko, Phys. Rev. B , 064403 (2008).24) E. Vargoz, P. Link, D. Jaccard, T. Le Bihan, and S. Heathman,Physica B , 225 (1997).25) P. Link and D. Jaccard, Physica B , 31 (1997).26) M. Nakashima, K. Tabata, A. Thamizhavel, T. C. Kobayashi,M. Hedo, Y. Uwatoko, K. Shimizu, R. Settai, and Y. ¯Onuki,J. Phys: Condens. Matter , L255 (2004).27) N. Kurita, H. Yamamoto, M. Hedo, T. Fujiwara, T. Shigeoka,S. W. Tozer, and Y. Uwatoko, Physica B , 1479 (2008).28) M. Nakashima, H. Kohara, A. Thamizhavel, T. D. Matsuda,Y. Haga, M. Hedo, Y. Uwatoko, R. Settai, and Y. ¯Onuki, J.Phys: Condens. Matter , 4539 (2005).29) T. Kagayama, T. Ishii, and G. Oomi, J. Alloys and Compounds , 263 (1994).30) G. Oomi and T. Kagayama, J. Phys. Soc. Jpn. , 2732 (1996).31) S. Raymond and D. Jaccard, Journal of Low TemperaturePhysics , 107 (2000).32) B. Cornut and B. Coqblin, Phys. Rev. B , 4541 (1972).33) Y. Nishida, A. Tsuruta, and K. Miyake, J. Phys. Soc. Jpn.75