Unconventional Strong Spin-Fluctuation Effects around the Critical Pressure of the Itinerant Ising-Type Ferromagnet URhAl

1.1 MB·English

by  Yusei Shimizu

Unconventional Strong Spin-Fluctuation Effects around the CriticalPressure ofthe Itinerant
Ising-TypeFerromagnet URhAl
Yusei Shimizu,1,∗ Daniel Braithwaite,1 Bernard Salce,1 Tristan Combier,1
Dai Aoki,1 Eduardo N. Hering,1 Scheilla M. Ramos,1 and Jacques Flouquet1
1Univ. Grenoble Alpes, INAC-SPSMS, CEA-Grenoble, F-38000 Grenoble, France.
(Dated:February3,2015)
Resistivity measurements were performed for the itinerant Ising-type ferromagnet URhAl at temperatures
downto40mKunder highpressureupto7.5GPa,usingsinglecrystals. Wefoundthatthecriticalpressure
5 oftheCurietemperatureexistsataroundPc ∼5.2GPa. NearPc,theA-coefficientoftheAT2 Fermi-liquid
1 resistivitytermbelowT∗islargelyenhancedwithamaximumaround5.2-5.5GPa.AbovePc,theexponentof
0 theresistivityρ(T)deviatesfrom2. AtPc,itiscloseton = 5/3, whichisexpectedbythetheoryofthree-
2 dimensional ferromagnetic spin fluctuations for a 2nd-order quantum-critical point (QCP). However, TC(P)
disappearsasa1st-orderphasetransition,andthecriticalbehaviorofresistivityinURhAlcannotbeexplained
b
bythetheoryofa2nd-orderQCP.The1st-ordernatureofthephasetransitionisweak,andthecriticalbehavior
e
isstilldominated bythespin fluctuationat low temperature. Withincreasing pressure, thenon-Fermi-liquid
F
behaviorisobservedinhigherfields.Magneticfieldstudiespointoutaferromagneticwingstructurewithatri-

2 criticalpoint(TCP)at∼4.8-4.9GPainURhAl.Oneopenpossibilityisthattheswitchfromtheferromagnetic
totheparamagneticstatesdoesnotoccursimplybutanintermediatestatearisesbelowtheTCPassuggested

l] theoreticallyrecently. Quitegenerally, ifadrasticFermi-surfacechangeoccursthroughPc,thenatureofthe
e interactionitselfmaychangeandleadtotheobservedunconventionalbehavior.
-
r
t PACSnumbers:71.27.+a,75.30.Kz,75.30.Mb,75.40.-s
s
.
t
a
I. INTRODUCTION plestotheorderparameter19,20. Theyshowedthata2nd-order
m
phase transition at high temperature changes to a 1st-order
- transitionbelowatri-criticalpoint(TCP)with1st-orderwing
d Since the 1960-1970s, the understandingof dynamic crit-
n ical phenomena and physical properties of itinerant spin- planes, which terminate at zero temperature in a finite mag-
o fluctuation systems has been one of the main topics in the netic field, i.e. at a quantum-critical-end point (QCEP)19,20.
c fields of magnetism in condensed matter physics1–6. This is Previously, it has also been discussed that the TCP emerges
[ due solely to the thermal spin fluctuations21,22 and the mag-
becausethesequestionsleadtounderstandnotonlyweakitin-
2 erantmagnetismind-andf-electronsystemsbutalsorecently netoelasticcoupling23,24.
v observed anomalous non-Fermi-liquid (NFL) behaviors and So far, the quantum criticality around the QCEP with the
1
magneticallymediated Cooperinstabilities7,8 caused by spin metamagnetic transition has been classified into the same
0
fluctuationsnearquantum-phasetransitions(QPTs). criticality as the QCP for an Ising-type transition25. How-
7
6 Sofar,itwaswidelybelievedthatbothitinerantferromag- ever, there is no symmetry change around a QCEP, whereas
0 netic (FM) and antiferromagnetic (AF) compounds usually the symmetry of the ordered phase is clearly distinguished
. have the quantum-criticalpoints (QCPs), where a 2nd-order from the paramagnetic (PM) phase for a QCP. It has re-
1
0 phase transition occurs at T = 0 by tuning some physical cently been pointed out theoretically that the quantum criti-
5 parameter,such as pressure, or atomic substitution, etc. The cality of the metamagnetic transition accompanied with the
1 self-consistent-renormalized(SCR)theorybyMoriyaandhis Fermi-surface change (Lifshitz transition) has another uni-
: coworkersgivesatheoreticalbasetodescribeNFLbehaviors versality class, which differs from other symmetry-breaking
v
i ofitinerantFM andAFmetallicsystemsnearQCPs6–8. Fur- phase transitions26,27. Also, as unconventionalsuperconduc-
X thermore, critical phenomena around magnetic QCPs were tivity associated with FM order has been discovered only
r investigated theoretically using the renormalization-group in uranium materials (UGe228, URhGe29, and UCoGe30), it
a
methodbyHertz9, andMillis10. Actually,someitinerantAF is intriguing to study the quantum criticality and the spin-
compounds obey the Moriya-Hertz-Millis theory for critical fluctuationeffectsaroundthe FM QPT for itineranturanium
behaviorsnearQCPs11. compounds.
However, for FM quantum criticality, the situation is dif- Recently, a FM wing structureand a QCEP have been re-
ferent. Surprisingly, it has been reported that an almost ported for UCoAl, which shows a 1st-order metamagnetic
FM helimagnet, MnSi12,13, and several ferromagnets, such transitionat∼0.7TwithaFMmomentof∼0.3µB/Uatlow
as UGe214,15, and ZrZn217,18, do not show the QCP at zero temperature31–33. This compound has a hexagonal ZrNiAl-
field but show a 1st-order phase transition when TC is sup- typestructurewithspacegroupP¯62m,inwhichthereisnoin-
pressedbyapplyingpressure. Toexplainthesebehaviors,re- versionsymmetry. Theuraniumatomsformaquasi-Kagome´
cently specific attentions were given to new quantum treat- lattice,thusmagneticfrustrationeffectsarepossiblyexpected.
mentfor FM QPT; for example, Belitz and Kirkpatrickcon- From high-pressure studies, it is considered that in UCoAl
sideredparticle-holeexcitationsfroma Fermisurfacewith a a TCP exists at negative pressure of −0.2 GPa34, and the
lowfrequencyandalongwavelength(softmode),whichcou- metamegnetictransitioncanbeexplainedbytheFMwings32.
2
d) II. EXPERIMENTALPROCEDURES
hifte (a) TC 4.15 GPa
s Single crystals of URhAl were grown by the Czochralski
y URhAl 4.3
nits.] (verticall 0 T 4.9 454...805 pmasthunueervlnelisidltniszcgreueelnsmolisdfseewt∼tirhvihtoi2htidy0ga0hoinnf×pisnraea1ssm0tiset0uputrrl×paee-srwae3(rs#0ecsruµe1frumpear-ne3ntrdu.afnoTc#eirhnm2.eg)esRdwdaeemhbvsiyiipccshluteeis3vwi9gin,et4eyg0roe.dmmWlieaeesemtasrmsyotuhnderadaiend---
u
b. 5.2 notallowaprecisedeterminationoftheformfactorofresis-
r
a tivity.Therefore,weextrapolatedA(P)linearlyto0GPa,and
R [ obtainedabsolutevaluesofρ(T,H)andAbynormalizingthe
8 12 16 20 24 28 extrapolatedvalue[A(P = 0)]tothezero-pressurevalue(A
T [K] =0.27µΩ cm/K2 forJ ⊥ c), sincethepressurevariationof
A-coefficientisalmostlinearforP <4.8GPa.
120 Thelow-T measurementswere carriedoutfor sample#1
(b) 5.53 GPa using a 3He-4He dilution refrigerator in temperatures down
to 40 mK and in fields up to 7 T under high pressures up to
m] 100 URhAl 5.23 6.03 7.5 GPa. Here, the magnetic field was applied almost along
c 0 T
Ω 6.63 the c-axis (easy-magnetizationaxis) and the current was ap-
ρµ [ 80 7.34 psulireedmpeenrtpseunnddiceurlhairgthoptrheessfiuerledwdeirreecptieornfo.rmTheedhaitgzhe-rTo mmaega--
neticfieldusing4Hecryostatforsample#1aswellas#2to
60 4.51 checkthereproducibility.
3.75
Asapressuretransmittingmedium,liquidargonwasused,
0.0 1.0 2.0 3.0 4.0 andpressurewasdeterminedwiththerubyfluorescencetech-
T [K] nique. For high-T measurements, since there is a volume
increase of helium gas in bellows of the pressure-tuningde-
FIG. 1: (Color online) (a) Temperature dependence of resistance viceaboveliquid-heliumtemperature,thismaycauseaslight
(verticallyshifted)ofURhAl(sample#1)between6and30K,mea-
change of the force, which is applied to the pressure cell.
suredatzerofieldandhighpressures,4.15,4.3,4.5,4.8,4.9,5.0,and
Then, the determinationof pressureis moreprecise for low-
5.2 GPa. The arrows indicate the Curietemperatures (TC) at each T measurementsbelow∼4Kthanforhigh-T measurements
pressure. Thedashedlinesareguidestotheeyes. (b)Temperature
above∼5K.
dependence of resistivity of URhAl (sample #1) below 4 K, mea-
sured atzero fieldand high pressures, 3.75, 4.51, 5.23, 55.3, 6.03,
6.63,and7.34GPa.
III. RESULTSANDDISCUSSION
In order to examine the pressure dependence of the Curie
temperatureofURhAl,wefirstshowthetemperaturedepen-
Since the TCP in UCoAl is estimated to exist at a negative denceoftheresistanceatvariouspressures(shiftedvertically)
pressure, it is notobservablefromhydrostatic-pressuremea- between6and30K inFig. 1(a). Onecanseetheclearkink
surements. In order to understand the critical phenomena anomaly in the resistivity curves due to the FM transition at
near the itinerant FM QPTs, further experimental examples the Curie temperature (TC), as indicated by the arrows [Fig.
arenecessary. 1(a)].TCshiftstolowertemperaturewithincreasingpressure,
andthekinkanomalybecomestoobroadtodetermineTCfor
Inthispaper,wereportpressure-inducedquantumcritical- P >5.0GPa.
ityofa5f itinerantIsing-typeFMcompound,URhAl,which Figure 1(b) shows results of resistivity measurements be-
has the same crystal structure as that of UCoAl. URhAl low 4 K at high pressures from 3.75 to 7.34 GPa. At 3.75
shows a FM transition at 25-27 K at ambient pressure36–38, and4.51GPa,TC is19and17K,respectively,andURhAlis
andtheFMmoment(∼0.9µB/U)isstronglyIsing-typewith FM in the temperaturerangeof Fig. 1(b) at these pressures.
themagnetization-easyaxisalongc,similartotheIsing-type The variation of resistivity ρ(T) is small at low temperature
metamagnetisminUCoAl. TheatomicradiusofRhislarger intheFM state. Ontheotherhand,from5.2to7.3GPa, the
thanthatofCo,sothe5f-electronicstateinURhAlmaycor- variation of resistivity is very large comparedto that at 3.75
respond to a state in which negative pressure is applied for and4.51GPa. Sincewedidnotobservethekinkanomalyin
UCoAl35. Therefore, the high-pressure study of critical be- theresistivityduetotheFMtransitionabove5.2GPa,URhAl
haviorsforURhAlcanhelptounderstandthemetamagnetism appearstobePMinthehigh-pressureregionabove5.2GPa.
inUCoAlaswellasthegeneralproblemofFMquantumcrit- Figure 2 shows the obtained T-P phase diagram at zero
icality. field. There is no significant sample dependence of TC(P).
3
25 TABLEI:TheA-coefficientofresistivity, A[µΩcm/K2], theelec-
URhAl tronic specific-heat coefficient γ [mJ/K2mol], and the values of
20 Pc 0 T A/γ2[µΩcm(molK)2/(mJ)2]forURhAl,UCoAl32,andUGe244.
Here,A(0)isthevalueofA-coefficientatambientpressureatzero
K] 15 FM PM field.
T [ 10 Curie Temperature P [GPa] A γ A/γ2 A/A(0)
URhAl 0(FM) 0.27 75 4.8×10−5 −
Sample #1
5 Sample #2 Pc ∼5.2 8 30
0 UCoAl 0 0.28 75 5.0×10−5 −
70 0.54 0.2 0.7
A
8
2] ρ PQCEP ∼1.5,7T 0.4 1.4
K 0 65 ρ
m/ 6 T * 0 [µ UGe2 0 0.007 30 7.8×10−6 −
Ω c 4 Sample #1 60 cΩ 1.3 0.1 110 8.3×10−6 14.3
µ m
A [ 2 55 ]
0 50 andA(P)arepredictedtovaryasT∗(P)∝(P −Pc)3/2and
3.0 A(P)∝(P−Pc)−1,respectively,leadingtoA∝(1/T∗)2/3;
K] inotherwords,A×(T∗)2/3 isconstant10,41. ForURhAl,we
* [ 2.0 obtain A ∼ 8 µΩcm/K2 and T∗ ∼ 0.4 K at Pc, leading to
T 1.0 A×(T∗)2/3 ∼4.3,andA∼3.5µΩcm/K2andT∗ ∼1.5Kat
0.0 7.5GPa,leadingtoA×(T∗)2/3 ∼4.6.Thisroughestimation
0 1 2 3 4 5 6 7 8
suggeststhattheobservedlargeA-coefficientemergesdueto
P [GPa]
the FM spin-fluctuation effects. However, we would like to
pointoutthepeculiarityofcriticalbehavioroftheFMQPTin
FIG.2: (Coloronline)T-PphasediagramofURhAlatzerofieldfor URhAl;asshowninthe3rdpanelofFig. 2,T∗(P)doesnot
thesamples#1and#2.Thedashedlinesareguidestotheeyes.The varyasT∗(P) ∝ (P −Pc)3/2 (thesolidcurve)anddoesnot
Aplo-cttoeedf.ficTihenetsaonliddtchuervreesiindduiaclarteessiTsti∗v(iPty)o∝fth(Pes−amPpcl)e3#/21,warheicahlsios go to zero as P → Pc. Also, the fact that T∗ is finite at Pc
predictedbythespin-fluctuationtheoryfora2nd-orderFMQCP10,41. conflictswithpresenceofa2nd-orderQCPinURhAl.
The maximum value of A ∼ 9 µΩcm/K2 in URhAl near
Pcisquitelargeforuraniumintermetalliccompounds.While
the heavy-electron superconductor UBe13 shows the excep-
TC(P)suddenlydisappearsabove5.0GPa. Ourresultssug- tionally large A-coefficient (∼ 90-100 µΩcm/K2)42,43, a lot
gestthattheFMcriticalpressureexistsatPc ∼5.2GPa. ofuraniumcompoundsshowtheA-coefficientoflessthan∼
InFig. 2,wealsoplottheA-coefficientandtheresidualre- 1 µΩ cm/K2 as summarized in the Kadowaki-Woodsplot43.
sistivityρ0asafunctionofpressure,whereρ(T)=ρ0+AT2. Table I shows the A-coefficient, the electronic specific-heat
ThepressurevariationoftheA-coefficientisverysmall(A∼ coefficient (γ), and the ratio A/γ2 for URhAl38, UCoAl32,
0.4-0.5µΩcm/K2)forP <4.8GPa,whereasitshowsadras- andUGe244. Besides,A/A(0)indicatestheratioofAdivided
ticincreaseabove5.0GPa.TheA-coefficientbecomesamax- by the A-coefficient at 0 GPa and zero field, i.e. A(0). As
imum(A∼9µΩcm/K2)ataround5.2-5.5GPa,suggestinga for UCoAl, the A-coefficient is A ∼ 0.28 µΩ cm/K2, and
large enhancement of the density of states at Fermi energy theelectronicspecific-heatcoefficientisγ ∼75mJ/K2mol32.
[D(ǫF)] of itinerant electrons above Pc. Interestingly, the TheA-coefficientofUCoAlincreasesneartheQCEP(∼1.5
largeincreaseoftheA-coefficient(A ∼4µΩcm/K2)beyond GPa, 7 T)32, but the enhancementof A is not so large com-
Pc indicatesthatthelargeenhancementofD(ǫF)remainsup pared to the pressure-inducedlarge A-coefficient in URhAl.
to∼7.5GPa(Fig.2). Thebehavioroftheresidualresistivity, Also, the A-coefficient of UGe2 increases ∼14-fold under
ρ0(P),accompaniesthebehavioroftheA-coefficient,A(P). highpressure,butthemaximumvalueofA-coefficientisnot
Below4.8GPa,ρ0(P)increaseswithincreasingpressureal- so large (∼ 0.1 µΩ cm/K2) at ∼1.3 GPa44. On the other
most linearly, then suddenly decreases with increasing pres- hand, a large A-coefficient (∼ 5 µΩcm/K2) near the critical
sureabove4.9GPa. At0Tρ0(P)showsastep-likebehavior pressurehasbeenreportedinanitinerantheavy-electronFM
at∼4.8GPaslightlybelowPc ∼5.2GPa. compound U4Ru7Ge645. The observed large A-coefficient
We also plot T∗, the maximum temperature for the T2- in URhAl near Pc is comparable with the value observed
regime, in the 3rd panel of Fig. 2. T∗ is about ∼ 2-3 K in in cerium heavy-electroncompoundssuch as CeCu2Si243,46.
thelow-pressureregionbelowPc,butitsuddenlydecreasesat Fromthecomparisonwithotherheavy-electronmaterialsus-
Pc, whereT∗(P)showsaminimum(T∗ ∼0.4K),andthen ingtheKadowaki-Woodsrelation,thequantumcriticalregion
graduallyincreaseswithincreasingpressure. inURhAlmaybedescribedgrossomodobythestronglycor-
InFMspinfluctuationframewitha2nd-orderQCP,T∗(P) related heavy quasiparticle with the large D(ǫF) caused by
4
85
10
5.53
100 (a)
(c) 6.63
80

90 URhAl 5.23 URhAl 8 URhAl
Ω cm]80 4.9 30 G TPa 6.03 6.63 7.34 Ω cm]7750 4 T 67..0334 5.53 2m/K] 6 01 TT
ρµ [70 4.82 ρµ [65 Ω c 4 23 TT
4.93 5.23 µ 5 T
60 3.75 60 A [ 7 T
2
50 55
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9
T 2 [K2] T 2 [K2] 0
90 75 70
(b) 6.03 6.63 (d)
µΩ cm]8700 U 2R ThAl 7.34 5 5.2.533 µΩ cm]7605 U 7R ThA 6l.03 55 6..52.6333 ρµΩ [ cm]0 665505
ρ [ ρ [
4.82 4.93 7.34
60 3.75 60 3.75 50
2 3 4 5 6 7 8
P [GPa]
50 55
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9
T 2 [K2] T 2 [K2] FIG.4:(Coloronline)PressuredependenceoftheA-coefficientand
64 (e) theresidualresistivityρ0 ofURhAl(sample#1)inzeroandmag-
62 URhAl 5.23 GPa neticfields,obtainedfromtheexpression,ρ(T) = ρ0+AT2. The

0 T dashedlinesareguidestotheeyes.
m] 60 T *
c
Ω 58
µ 6.03
ρ [ 56 T * for0Tmeasuredat5.23,6.03,and7.34GPaasafunctionof
54 7.34 T2. The arrows indicate the temperature, T∗ (Fig.2), below
52 whichthe T2-regimeworks. Foran appliedfield of 2 T, the
0.0 0.2 0.4 0.6 0.8 1.0
T 2 [K2] large A-coefficient is suppressed at 4.93, and 5.23 GPa, and
ρ(T) shows the T2-temperature dependence, similar to that
FIG.3: (Coloronline)ρ(T)vsT2 plotofURhAl(sample#1)at at4.82,and 3.75GPa. On theotherhand, the slopeof ρ(T)
highpressuresfrom3.75to7.34GPain(a)0T,(b)2T,(c)4T,and becomeslargeat around6-6.6 GPa at 2 T. For 4 T, and 7 T,
(d)7T.(e)Theenlargedfigureofρ(T)curvesfor0Tmeasuredat thevariationofρ(T)athigh-pressureregionabove∼6.6GPa
5.23,6.03,and7.34GPaasafunctionofT2below1K.Thearrows becomeslargerthanforpressuresbelow∼6.0GPa.
indicateT∗(Fig.2),belowwhichtheT2-regimeworks. Here,T∗at
InFig. 4,wesummarizethepressuredependenceoftheA-
7.34GPaisabout∼1.1K.Thesolidlinesaretheresultsoffitting
byρ(T)=ρ0+AT2. c#o1e)f,ficoibetnatinaenddftrhoemretshiedueaxlprreessissitoivni,tyρ(ρT0)of=UρR0hA+lA(sTam2,pilne
zeroandmagneticfields. Withincreasingmagneticfield,the
divergence of the A-coefficient is suppressed, and the step-
spin fluctuations. However, we should be careful about the
abovediscussionsincethevalueofA/γ2 isnotuniversalde- likebehaviorofρ0becomesbroad(Fig. 4).
pendingonthecorrelationofthesystem47–49. Since the behaviorsof the A-coefficientand ρ0 above 5.0
Next,weshallseelow-T ρ(T)curvesinzerofieldandmag- GPa differevidentlyfromthose below∼ 5.0GPa inthe FM
neticfields. Figures3(a),(b),(c),and(d)showtheresistivity state, it is considered that URhAl is not in the FM state any
ρ(T) vs T2 under high pressures from 3.75 to 7.34 GPa for moreabove5.0GPa. Thisisconsistentwiththefactthatthe
0, 2, 4, and 7 T, respectively. For zero field, at lower pres- anomaly due to the FM transition disappearsabove 5.0 GPa
sures than 4.8 GPa, we find ρ(T) = ρ0 +AT2 behavior, as (Fig. 2). TheCurietemperature,TC(P),possiblybecomesa
predictedfor an itinerantFM state at low temperature(T ≪ 1st-orderphasetransition,andthenTC(P)suddenlycollapses
TC)50. Ontheotherhand,theresistivityshowsaremarkable above5.0GPa.
variationwith a largeincreaseof the slope(A-coefficient)at TheexperimentallyobservedlargeenhancementoftheA-
0 T between 4.82 and 4.93 GPa. Around 5-6 GPa, the tem- coefficient suggests a large mass enhancement due to spin-
peratureregionwherethe resistivitycanobeythe expression fluctuation effects and/or a variation of Fermi surface. Gen-
ρ(T) = ρ0 + AT2 is much smaller than at 4.82, and 3.75 erally, large fluctuation effects occur for a 2nd-order phase
GPa. InFig. 3(e),weshowtheenlargedfigureofρ(T)curves transition with a divergence of the correlation length of the
5
thenalargemaximumintheA-coefficientmayemergedueto
10
theincreaseofcorrelationlengthasT →0.
4.51 GPa
URhAl 4.93 GPa Figure5showstheA-coefficientandtheresidualresistivity
8
5.23 GPa as a function of magnetic field. At 4.5 GPa, A(H) value is
2K] 5.45 GPa verysmall,andA(H)monotonicallydecreaseswithincreas-
m/ 6 56..5033 GGPPaa ingfield. At5.0GPa, theA-coefficientbeginsto increasein
Ω c 6.90 GPa zeroandlowfields,andA(H)issuddenlysuppressedbymag-
µ 4 7.34 GPa neticfieldof∼1T.Ataround5.2-5.5GPa,theA-coefficient
A [ isverylargeinzerofieldandremainslargeupto1-1.5T,then
2 rapidlydecreasesathighfields(1.5-2T).At6.0GPa,thede-
creaseofA(H)occursathigherfieldnear3T.At6.9and7.3
GPa,thevalueofA-coefficientat0Tbecomesabouthalfof
0
theAvalueat5.5GPa,andaftershowingaslightmaximumat
around2 T, itmonotonicallydecreaseswith increasingfield.
m] 64 At6.9and7.3GPa,ρ0(H)increaseswithincreasingfield,and
c showsasmoothmaximumataround3T.
Ω 60 To search for the FM wing structure, we look at the mag-
µ
[0 56 netic field dependence of the resistivity [ρ(H)] under high
ρ pressure. Figure 6 showsρ(H) under5.53GPa at 2.5, 1.75,
52 1, 0.75, and 0.1 K with A(H) and ρ0(H) obtainedfrom the
temperaturedependenceof ρ(T) = ρ0 +AT2. ρ(H) curve
0 1 2 3 4 5 6 7 8
bendsataround2.5-3Tforeachtemperature. Wedefinethe
µ H [T]
0 anomalyatHmatT →0fromA(H)andρ0(H)asindicated
bythearrowsinFig. 6.Atthelow-fieldregionbelowHm,the
FIG. 5: (Color online) Magnetic-field dependence of the A- A-coefficientis very large comparedto the high-fieldregion
coefficient and the residual resistivity ρ0 of URhAl (sample #1), above Hm. On the other hand, the high-field region above
obtainedfromtheexpression,ρ(T)=ρ0+AT2. Hm correspondsto the FM side, where the resistivity obeys
90 ρ(T)=ρ0+AT2withthesmallA-coefficient.
Theanomalyinρ(H)curvessupportsthepresenceofFM
2.5 K
80 wing structure in URhAl. In Fig. 7(a), we plot the P-H
1.75 K
m] phase diagram for Hm obtained from the magnetic field de-
µΩ c 70 1 K pµe0nddHemnc/edsPof∼A3-c.5oe±ffi0c.1ienTt/GanPda.ρT0.henWweeobestatiimnattheethreelaTtiCoPn
ρ [ 60 at PTCP ∼ 4.8-4.9GPa, when Hm → 0, using the value of
0.1 K µ0dHm/dP. In UCoAl, which has the same crystal struc-
50 0.75 K URhAl ture as URhAl, a clear 1st-order metamagnetic transition is
8 5.53 GPa 70 seenatHm duetotheFMwing. Accordingtohighpressure
studies32, the metamagnetic transition field of UCoAl varies
2K] 6 65 asµ0dHm/dP ∼3.5T/GPa32,whichisverysimilartothatof
µΩA [ cm/ 4 Aρ 60 [ cmρµΩ UinRThaAblle..WIIe. summarizethevaluesofPTCP andµ0dHm/dP
0 ]
2 55
TABLEII:ThevaluesofPTCP andtheslopeoftheFMwing, i.e.
µ0dHm/dP forURhAlandUCoAl32.
0 50
0 1 2 3 4 5 6 7 8 PTCP[GPa] µ0dHm/dP [T/GPa]
µ0H [T] URhAl 4.8-4.9 3.5±0.1
UCoAl −0.2 3.5
FIG. 6: (Color online)Magnetic field dependence of resistivity of
URhAlforthesample#1,measuredat5.53GPa. Thedashedlines
areguidestotheeyes. WhenwecrosstheFMwing,weexpectthe1st-orderPM-
FM phase transition at Hm. At the 1st-order transition in
UCoAl,theA-coefficientshowsastep-likebehaviorasafunc-
tionofmagneticfield32. Ontheotherhand,thestep-likebe-
magnetic order. In contrast, such an effectwould not be ex- haviorintheA-coefficientofURhAlnearHmisratherbroad.
pected for a 1st-order phase transition. Nevertheless, if the ThismayindicatethatthetransitionatHmisweakly1st-order
transitionnearthePcisonlyweakly1st-orderandthedropof in URhAl. However,the samplequalitycan be the originof
theFMmomentattheFM-PMphasetransitionisverysmall, thebroadnessofthetransition.
the critical behavior becomes similar to that of a QCP, and We shall compare A(H) for URhAl with that for UCoAl.
6
T
Curie Temperature
T
7 (a) C
P ~ 4.8-4.9 GPa
TCP
URhAl URhAl
2nd Order
PM Weakly 1st Order
6
a] TCP
P QCEP
G
5
P [ from ρ (H) 0 GPa
0 P
FM (0 T) H
from A(H)
4 QCEP Pc ~ 5.2 GPa
µdH /dP ~ 3.5 T/GPa
0 m
3
0 1 2 3 4 5 6 7 FIG.8: (Coloronline)SchematicT-P-H phasediagramoftheFM
µ H [T] wingsinURhAl[SeealsothetoppanelofFig.2andFig.7(a)].
0
A [µΩ cm/K2]
2 4 6 8 at ∼ 5.0 GPa is about20 times larger than the A-coefficient
above Hm (Fig. 5). In UCoAl, on the other hand, the A-
coefficient in the PM state below Hm is about only 2 times
7 4 largerthantheA-coefficientaboveHm at0and0.54GPa32.
The difference of the A-coefficient between URhAl and
6 6 5 3 1 2 UmCenotAs;lthmeaFyMbeorrdeelareteddmtoomtheanttoinfUthRehmAalgisne3titcimoersdelarregderm(o∼-
Pa] 7 0.9 µB/U ) than the magnetic-field-inducedFM moment (∼
G 5 (b) 0.3 µB/U) in UCoAl. As shown theoretically by Yamada21,
P [ URhAl thermallyfluctuatingmagneticmomentsenhancethescatter-
ing of quasiparticles in the PM state, and may cause such a
4 largeA-coefficient. At present, theorderedmomentnearPc
hasnotyetbeenstudiedforURhAl,sofurtherstudiesarenec-
essarytoclarifythispoint.
3
The behavior of the A-coefficient changes in association
0 1 2 3 4 5 6 7 with the wing structure. To see the relationship between
µ H [T]
0 the enhancementof A-coefficientand the FM wing, it is in-
triguingto see the contourplotof A-coefficienton the P-H
FIG.7: (Coloronline)(a)PlotoftheobservedanomaliesinA(H) phasediagram. Figure7(b)showsthecontourplotoftheA-
andρ(H)ofURhAlontheP-H phasediagramforthesample#1. coefficientontheP-H phasediagram,obtainedfromρ(T)=
Thedashedlineindicatestheresultoflinerfitting. (b)Contourplot ρ0+AT2. Thered-coloredregioninthisplotshowstheen-
oftheA-coefficientofresistivityonURhAlontheP-H phasedia-
hancement of the A-coefficient, whereas the purple-colored
gram,obtainedfromtheexpressionρ(T)=ρ0+AT2forthesample region shows the small A-coefficient. The A-coefficient is
#1.
largest at around 5.2-5.5 GPa in zero field. With increasing
pressure and magnetic field, the A-coefficientis suppressed.
ThelargeenhancementofA-coefficientoccursoutsideofthe
ForUCoAl,astep-likebehaviorinA(H)atHmisseenunder FMwing(redregion).
lowpressureregionbelow0.54GPa.For0.54GPa,thediffer- InFig. 8,weplottheschematicT-P-H phasediagramof
enceofpressurefromtheTCP(∼−0.2GPa)isestimatedto the FM wings in URhAl [See also the top panel of Fig. 2
beδP ≡P −PTCP ∼0.74GPa. SinceweestimatePTCP ∼ and Fig. 7(a)]. The theoretically suggested 1st-order wing
4.8GPa forURhAlin thepresentwork,thepressureof0.54 planes terminate at the QCEP at zero temperature in a finite
GPa inUCoAlmaycorrespondto apressureof4.8+δP ∼ field. InUCoAlthe magnetic-fielddependenceofA(H) co-
5.54GPa in URhAl. At∼ 5.5GPa, we obtainHm ∼ 2.5 T efficient shows a sharp maximum at the PM-FM transition
forURhAlfromFig. 7(a),whichisclosetothevalueofHm near the QCEP (P ∼ 1.5 GPa in H ∼7 T)32,38. Such a
forUCoAlat0.54GPa. field enhancementin A(H) was not observed in the present
Incontrast,theenhancementoftheA-coefficientinURhAl work, suggesting that the QCEP of URhAl may exist above
is much larger than the value in UCoAl (see, Table I), sug- ∼7T.Alternatively,theinterplaybetweenspinfluctuationand
gesting that the density of states (mass enhancement and/or Fermi-surface instability can lead to complex phenomenaas
thechangeofFermisurface)neartheQPTismoredrasticin discussedlater.
URhAlthanUCoAl. InURhAl,theA-coefficientbelowHm Asclosetothe criticalpressure,thetemperaturerangefor
7
at Pc or if there is a mark of a new pressure-induced phase
2.4 intermediatebetweentheFMandthePMphases.
URhAl
0 T
Recently it has been shown theoretically that anothernew
2 T
2.2 phase possiblystabilizes near the FM-PM QPT51–56; if there
7 T
are quantum fluctuations in terms of fermionic particle-hole
n 2.0 excitations on the Fermi surface, some deformations of the
Fermi surface enhance the phase space available for the
1.8
quantum fluctuations, and such a Fermi-surface instability
causesanothertypeoforderingtogainthefreeenergyofthe
1.6
system53–56.
2 3 4 5 6 7 8 9 Ithasbeenshownthattwocasesofneworderingarepos-
P [GPa]
sibleneartheFM-PMQPT:(i)spiralmagneticphase,and(ii)
spin-nematic phase54. The energy scale of the spin-nematic
FIG.9: (Coloronline)Pressuredependence oftheexponent(n) of phasetransitionisalmost10timessmallerthanthecaseofspi-
resistivity[ρ(T)=ρ′0+A′Tn]forURhAl(sample#1)at0,2,and ralmagneticphase54,thereforeaspiralmagneticphasemight
7T.Thedashedlinesareguidestotheeyes. bemorelikelytooccur. Aspiralmagneticphaseemergesbe-
lowtheTCPasan intermediatestatebetweentheuniformFM
andthePMstates54,55.ThetransitionbetweentheuniformFM
Fermi-liquidregime[ρ(T) = ρ0+AT2]isfoundtobevery andthe spiralmagneticstates occursas a Lifshitz transition,
small(T∗ ∼ 0.4K),an alternativedescriptionis tofocuson whereas the transition between the spiral magnetic state and
theNFLbehavior. Weanalyzedtheresistivitydatabytheex- the PM state is of 1st-order54. Interestingly, an anisotropic
pression,ρ(T)=ρ′0+A′Tn.Here,themaximumtemperature dispersionforelectronbandchangesthenatureofthespiral-
fortheNFLregimeρ(T) = ρ′0+A′Tn,i.e.,T∗∗ ∼ 2.2Kat to-PM phase transition, and the transition possibly becomes
∼ Pc. Figure 9 shows the pressuredependenceof the expo- 2nd-order54.
nentofresistivity(n). AsseeninFig. 9,at0Ttheexponent Thepossiblepresenceoftheintermediatenewphasemight
n is about2 below 4.8 GPa, whereasn(P) decreases with a explainwhywecannotseeaclear1st-ordertransitionbetween
step-likevariationtoaminimumataround∼5.0-6.0GPa. At the FM and the PM states above Pc, differentfrom the case
around ∼5.0-6.0GPa, the values of n are about 1.6-1.8. At of UCoAl. In order to explore such a new phase around the
2 T and 7 T, the step-like behavior and the minimum of the QPT,furtherexperimentalstudiesarerequired. Inparticular,
exponentn(P) shift to higher pressure regions. At 7 T, the measurements of thermodynamic quantities and observation
dipofn(P)ataround∼7.0GPabecomesshallowandbroad, of Fermi-surface change through Pc for URhAl, though ex-
andthevalueofn(P)becomesslightlylargerthanat0Tand perimentallychallenging,woulddeepentheunderstandingthe
2T. Fermi-surfaceinstabilityandthenatureoftheFMQPT.
The exponent n (Fig. 9) varies as a function of pressure InURhAl,nosuperconductivitywasobservedunderhigh-
corresponding to the behavior of the A-coefficient (Fig. 4). pressureupto7.5GPa atlow temperaturedownto∼ 0.1K.
Above Pc, the exponent n is nearly 1.6-1.7, which is close At presentwe cannotrule outthe possibilitythat the sample
to the value (n = 5/3) suggested from the SCR theory for qualityaffectstheemergenceofsuperconductivityandthesu-
three-dimensionalFMspinfluctuationnearaQCP.However, perconductingtransitiontemperatureistoolowtobedetected.
thisNFL behaviorseemstoconflictwith thepresenceofthe However,evenifasuperconductivitydoesnotoccur,theFM
FM 1st-order wing structure in URhAl. A weakly 1st-order systemwouldresolvetheinstabilityduetothequantumfluc-
nature at the FM wing may explainthe NFL behaviorin the tuationatT →0bytheoccurrenceofanothernewphaseas-
resistivity. sociated with the Fermi-surfaceinstability as mentionedjust
Hereafter, we note that the critical behavior around Pc in above.
URhAlisdifferentfromthetheoreticalsuggestionfora2nd- It is interesting to consider why the intermediate phase
order FM QCP. It is suggested that the ratio of T∗ for the possibly appears in URhAl. One may consider that the
Fermi-liquid regime to T∗∗ for the NFL regime is enhanced lack of local inversion center and/or the quasi-Kagome lat-
as T∗∗/T∗ ∝ (P − Pc)−3/4 as approaching Pc for a 2nd- tice in ZrNiAl-type structure can induce the intermediate
orderFMQCP10,41. ForURhAl,T∗∗doesnotchangeclearly, phase. However,theZrNiAl-typehexagonalsymmetrystruc-
i.e., T∗∗ ∼ 2.1±0.2 K. Inaddition,T∗(P) is almostlinear ture (P¯62m: D3 ) does not lead to Dzyaloshinskii-Moriya
3h
(Fig. 2). These experimentalresults forURhAl suggestthat interaction57, which could induce a helimagnetic order58.
thespin-fluctuationeffectscannotbeexplainedsimplybythe Such an intermediate phase has not yet been observed in
2nd-orderFMQCPagain. UCoAl,whichhasthesameZrNiAl-typestructure,aroundthe
InURhAl,theNFLproperties(seeFig,9)areobservedfar PM-FMphasetransitioninducedbyuniaxialstressalongthe
above Pc, and the pressure domain of the enhancement of c-axis59–61. Therelationshipbetweenthecrystalstructureand
the A-coefficient appears quite asymmetric around Pc. Fur- theoccurrenceoftheintermediatephaseremainsopenques-
thermore the enhancement of the A-coefficient extends over tion.TheauthorsinRef.54suggestthattheintermediatephase
a large P window (5.5-7 GPa). Then the key question is if maygenerallyoccurevenforasimplesphericalFermisurface
the switch from the FM state to the PM state simply occurs due to the Fermi-surface instability accompanyingthe quan-
8
tum fluctuations (particle-hole excitations on the Fermi sur- andthemagnetic-fielddependencesoftheresistivity maybe
face). As seen in Fig. 9, the NFL behaviorof the resistivity consistentwith the presenceof a FM wing structure with an
isremarkableinURhAlfarabovePc. Suchstrongquantum- estimatedTCPat4.8-4.9GPa. Atleastwiththepresentqual-
fluctuation effects near the FM-PM QPT may invoke the in- ityofthecrystalthe1st-orderphasetransitionappearsweak.
termediatephaseinthismaterial. TheresistivityshowstheNFLbehaviorabove5.0GPaupto
7.5GPa. URhAlmaybeamaterialinwhichtheswitchfrom
the FM state to the PM state occursthroughan intermediate
IV. CONCLUSION phasearoundtheQPT.
The quantum criticality of the three-dimensional-Ising-
type itinerant FM compound URhAl was studied by low-
ACKNOWLEDGMENT
temperatureresistivity measurementsunderhighpressureup
to7.5GPa.TheCurietemperatureissuppressedwithincreas-
ing pressure, and suddenly disappears above 5.0 GPa. Our WewouldliketothankS.Kambe,G.Knebel,K.Ishida,Y.
resistivity results suggest the FM critical pressure of ∼ 5.2 Tada, K. Hattori, S. Hoshino, and Y. Ikeda for valuable dis-
GPa. Above5.2GPa,thegroundstateisnotFM,andtheA- cussions and helpful comments. This work was supported
coefficientislargelyenhancedataround5.2-5.5GPainzero by ERC starting grant New Heavy Fermion, KAKENHI,
and low-field region. The characteristics of the temperature REIMEI,ICC-IMR,andANRprojectPRINCESS.
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