Table Of ContentThermoelectric transport across the metamagnetic transition of CeRu Si
2 2
Heike Pfau, Ramzy Daou, Manuel Brando, and Frank Steglich
Max Planck Institute for Chemical Physics of Solids, No¨thnitzer Str. 40, 01187 Dresden, Germany
(Dated: January 18, 2012)
We have measured the thermopower across the metamagnetic transition of the heavy fermion
compoundCeRu2Si2attemperaturesdownto0.1Kandmagneticfieldsupto11.5T.Wefindalarge
negative enhancement of the thermopower on crossing the metamagnetic field, as well as a sudden
changein slope. Wearguethat thisis consistent with theZeeman-drivendeformation of theFermi
surface through a topological transition. The field dependence of the thermopower highlights the
discrepancy between thermodynamicand transport properties across themetamagnetic transition.
2
1
PACSnumbers: 71.27.+a,72.15.Gd,75.30.Kz,72.15.Jf
0
2
n I. INTRODUCTION spin-splitting and considerable spin-dependence of the
a quasiparticlemasseswasdirectlydetectableviadeHaas-
J
In heavy fermion systems, the weak hybridisation be- vanAlphen(dHvA) measurementsinsome ofthe lighter
7 bands14,15.
tween nearly localized f-electrons and the mobile con-
1
How then, can we explain relatively sharp features in
duction electrons leads to a Fermi liquid ground state
transportapparentlycoupledtosmoothbehaviourinthe
] with narrow bands and quasiparticles with strongly en-
l thermodynamics? Why do they occur at the same mag-
e hanced effective electronic masses. The heavy fermion
- bandwidth can become comparable to the Zeeman en- netic field? To address these questions we have carried
r
outahighresolutionstudyofthethermopowerandther-
t ergy in accessible magnetic fields. The Fermi surface
s mal conductivity of CeRu Si in magnetic fields up to
. can therefore be significantly deformed as the bands are 2 2
t 11.5T and to temperatures down to 0.1K. The ther-
a spin-split by an applied magnetic field, and topological
m (or Lifshitz1) transitions are possible. These transitions mopowerisaffectedbyboththescatteringprocessesand
thermodynamicpropertiesofthe electronfluid, andsois
- break no symmetry and appear thermodynamically as
d crossovers at any finite temperature2. They are, how- anideal probe to investigate the connection between the
n two. It is also predicted to acquire a singular part close
ever, true quantum phase transitions in the limit of low
o to a Lifshitz transition2.
temperatureandarethereforesurroundedbyaregimeof
c
In this paper we show that the low temperature be-
[ quantum critical fluctuations. There has been renewed
interest recently in the possible connection between Lif- haviour of the thermal transport properties of CeRu2Si2
2 arealsocompatible with aLifshitz transitionmodel. We
shitztransitionsandquantumcriticalbehaviourinheavy
v find multiple sign changes in the field dependent ther-
fermion and strongly correlated materials as an alter-
0
mopower, as well as a large negative peak at the MMT
native to scenarios which involve competition with the
6
1 Kondo mechanism3–11. that persists in the low temperature limit. There is a
sharp kink in the thermopower precisely at the metam-
1 In this context the heavy fermion paramagnet
. CeRu Si has attracted much interest. For a good re- agneticfield,similartothatseenintheHalleffect. Using
0 2 2
simplemodelswecanreproduceeitherthesharpfeatures
1 view of the many thermodynamic, transport and spec-
in transport, or the peak in γ, but not both simultane-
1 troscopic experiments on CeRu2Si2, see Ref. 16. When
1 a magnetic field of ∼ 7.8T is applied along the crys- ously. Weconcludethatsingle-particlemodelswithrigid
: talline c-axis,allthermodynamicpropertiesshowa large band shifts are alone insufficient to describe the MMT
v
and must be extended.
i anomaly, the most obvious of which are a rapid non-
X
linear rise in the magnetisation and a pronounced peak
r intheelectronicspecificheat,γ. Thethermodynamicsof
a II. EXPERIMENTAL
this metamagnetic transition(MMT) areconsistentwith
proximitytoaquantumcriticalendpoint12,13. Inthelow
A. Methods
temperature limit, however, the material exhibits Fermi
liquidlikepropertiesatallmagneticfieldsandtheMMT
is always a crossover. The phase diagram is uncompli- The sample of CeRu Si used here was grown by
2 2
catedby orderedphases suchas magnetismorsupercon- F.S. Tautz in Cambridge. The residual resistivity, ρ ,
0
ductivity. Meanwhile, low temperature Hall effect mea- was 1.1µΩcm, compared to 0.4µΩcm for a sample from
surements show a very small but distinct anomaly that the same batch measured in Ref. 5. Thermal transport
appears to be sharper than the thermodynamic signa- properties were measured on a single crystal of dimen-
tures ofthe MMT5, but occurs atthe same criticalmag- sions 3.5×0.28×0.08mm3 (a×b×c) using the standard
netic field. This was interpreted as a sign of a topolog- one-heater, two-thermometer technique. Low resistance
ical transition of the Fermi surface, driven by enhanced contacts were made by soldering to the sample. A sys-
Zeeman splitting of the heavy fermion bands. Indeed, tematicuncertaintyofabout10%thatreflectsthesizeof
2
the contacts is ignored here as it acts as a scaling factor
only. Magnetic field was applied along the crystalline c- 2) Daou (2006) a)
uatainhxgseeeitsdh,mwewaehsahaoslbiteul-heprefieltamehrnleedeefrne.matr,enaSnrdluecapsetuneelldrmetciaopnednlgeesdrciaunfttorcuirbtcriaanetchlgkrecgaNurntrobhgruTeeenrnimsdttfis-ouflrewadelmieeeercdedetn.rataitcspaTpvwholovieeelrsetdeer- µΩoefficient (cm/K 1234 Axoo,kyxwi ss(n1cc 9daa9allee8tadd) 000...123 2242C/T) (J/Kmol)
filaments also permitted in-situ four-point d.c. resistiv- A c BMMT=7.8T (
irteyvemrseeadsutroecmheenctks.foTrhtheecrumroreenletctarnicdcmonatgrnibetuitciofinesldtowtehree 03 453700mmKK b) 0
230mK
d.c. resistivity, but these were found to be negligible for m) 2.5 100mK
thecurrentsemployed(10<I <150µA).Thiswasalsoa Ωc
µ 2
utiseesfu(il.ech.eHckaltl,otchoenrfimraml Hthaaltl atrnadnsNveerrnsesttreaffnescptso)rtmpardoepenro- ρ (xx 1.5
detectable contribution to the measurements. Temper-
1
ature sweeps were performed at several fixed magnetic
fields up to 11.5T. Additionally, magnetic field sweeps Daou (2006) 100mK c)
T) 0.6 x scaled
were perfomed at several fixed temperatures. In order m/ Daou (2006) 15mK
to avoid the high thermal resistance of superconducting µΩc 0.4 own datax 1s0c0amleKd
contacts used in the experimental stage, the minimum B (
field we applied was 0.2T. ρ/dxx 0.2
d
0
1.1
B. Results 2.4 d)
m) 2 L/L0 1 120360mmKK
1. Magnetoresistance 2K 445mK
W/ 1.6 0.9 570mK
T ( 0 4 8 12
We beginbycomparingthe magnetoresistancedatato κ/ 1.2
previously available data from Ref. 5. As in previous
0.8
studies, at low temperatures the resistivity (ρ) is Fermi-
liquid like, i.e. ρ = ρ +AT2, at all fields. Figures 1a) 1 e)
0
and b) show A and ρ extracted from temperature and 0.5
field sweepsrespectively. Ais proportionalto the square 2K) 0
othfethKeaedloecwtarkoni-iWcospoedcsifireclahteiaotnschoiepffi18ciaenndt,iγs,haecnccoerdainthgetro- µS/T (V/ -0-.15 120360mmKK
-1.5
modynamic property. This relationshiphas been seento 445mK
hold well in other samples19,20. Clearly the width of the -2 570mK
peak in A does not have a strong sample dependence. 0 2 4 6 8 10 12
Looking at dρ/dB in Fig. 1c), however, we can see that Magnetic Field (T)
the peak at the MMT is much narrower for the sam-
ple in Ref. 5. Small differences in sample quality affect
FIG.1: (Color online)Lowtemperaturetransportproperties
ρ , which arises only from electron-impurity scattering,
0 of CeRu2Si2 as a function of magnetic field, B. a) The A-
much more than thermodynamic properties.
coefficientoftheresistivity,ρ,peaksatBMMT =7.80±0.05T.
Examining the data from Ref. 5 more carefully, we
For comparison, similar data is included for the sample of
see that dρ/dB peaks at precisely the same field as that
Ref. 5 and also published specific heat data on another sam-
wherethesmallnegativedipintheHallresistivityoccurs. ple17. b) A peak in ρ(B) appears at higher temperatures
In the current data, the peak in dρ/dB is at BMMT = (lighter curves). The resistivity measured at 100mK is close
7.80±0.05T.ThiscoincideswiththepeakinAextracted to the residual resistivity ρ0, however, and shows no such
from the temperature dependence of the resistivity at peak at the MMT. c) The derivative of ρ0 with respect to
fixed fields, within the resolution of the measurement. magnetic field, compared to the sample of Ref. 5. The data
of Ref. 5 presented in a) and c) has been slightly scaled so
Other than the ordinary magnetoresistance arising
from cyclotron motion of the electrons, variation of ρ that BMMT matches, accounting for a misalignment of the
0 sample. TheMMT appearsmuchbroaderinρofthepresent
with magnetic field can arise either from changes in the
sample. d)Thethermalconductivity. Fieldsweeps(filledcir-
shapeoftheFermisurface(orbitalcontributions)orfrom
cles) and temperature sweeps (open squares) are in excellent
modification of the impurity potential, perhaps due to
agreement. The inset shows the Lorenz ratio at 100mK. e)
enhanced zero-temperature fluctuations21. Calculations Thethermopowerdividedbytemperatureshowsmultiplesign
based on this second mechanism in the vicinity of a fer- changes and a strong negative peak near the MMT. There is
romagneticquantumcriticalpointshowthatthefluctua- asharpkinkinthethermopowerpreciselyatBMMT whichis
moreclearlyseeninFig.2. Anadditionalpeakat1Tbecomes
sharper as thetemperature is reduced.
3
tionsgenerateapeakinρ atthe criticalfield. We know have electron-like Fermi surfaces. This sign inversion is
0
that the ferromagnetic fluctuations in CeRu Si are also anexpected consequenceofthe heavyfermionstate27,28.
2 2
peaked around the MMT22. ρ shows a monotonic in-
0 In the particular case of CeRu Si , the positive Hall
2 2
crease, however, suggesting that the orbital mechanism
effectsuggeststhathole-likecarriersdominatethetrans-
is more important.
port at all fields. The positive S/T at B = 0 implies
conversely that it is electron-like carriers that dominate.
If we assume that the large hole-like sheet of the Fermi
2. Thermal Conductivity surfacewiththemostenhancedeffectivemassdominates
the charge transport, we would expect a negative ther-
The thermal conductivity, κ, is shown in Figure 1d. mopower. In a multiband system, however, the sum
When only electronic transport is involved, it is di- of contributions to the thermopower is weighted by the
rectly related to the electrical conductivity, σ, via bandconductivity,andS canacquireeithersigndepend-
the Wiedemann-Franz law (WFL), which is written as ing on the details of the bandstructure. The positive
κ/σT = L where L = π2kB2 is Sommerfeld’s constant. ‘background’ S/T at low and high fields would be a re-
0 0 3e2 sult of this. We could then interpret the large negative
This ratio holds to within 5% at all fields at 100mK, as
peak in S(B)/T as an enhancement of the contribution
we might expect in the Fermi liquid regime (see inset to
of a hole-like surface with significant f-character that is
Fig. 1d). Previous studies down to 0.64K were not able
tuned with magnetic field, with a maximum effect near
to test the validity of the WFL in the low temperature
B . The naturalcandidate for this is the heavy hole
limit23,24. MMT
band that has been observed by dHvA.
b. q-ratio AnempiricalrelationshipbetweenS/T as
T →0 and γ has been established in heavy fermion sys-
3. Thermoelectric power
tems26. This relationship is quantified by the q-ratio,
where q = S NAve ≈ ±1. N is Avogadro’s number.
The thermopower,S,ofCeRu Si hasa complextem- T γ Av
2 2 It is based on a series of approximations; in particular
perature dependence reflecting the compensated multi-
that the derivative of the electrical conductivity, σ, in
band electronic structure23. There are multiple sign
the Mott formula, S/T =L e(∂lnσ/∂ε) , can be writ-
changes in the temperature range below 10K. At low 0 εF
ten as (∂lnτ(ε)/∂ε+∂lnN(ε)/∂ε) ≈ 1/ε . N(ε) is
temperature and low fields S/T reaches a value of εF F
0.7µV K−2, considerably smaller than that reported by the electronic density of states and τ is the scattering
lifetime.
Ref. 23. This difference may be the result of the small
0.2T field that was the lowest field that we applied, or InFig.2weshowγ derivedfromtheA-coefficient(γA)
it may reflect the dependence of the thermopower in a via the Kadowaki-Woods relationship and compare it to
compensated multiband metal on sample quality. thatderivedfromS/T viatheq-ratio(γS). Theenhance-
Figure 1e) shows that S(B)/T changes sign twice at mentofγA atthe MMTcomparedtothezerofieldvalue
low temperature as the magnetic field increases. There is similar in magnitude to the enhancement of γS. The
isa broadnegativepeakinthe thermopower,apparently detailed field dependence is very different, however. The
at a field slightly below B . There is an additional peak at 1T is not correlated with any feature seen in
MMT
peak at 1T that becomes more sharply defined at low other thermodynamic or transport properties. The kink
temperatures which is not associated with any known inS/T atBMMT correspondstothelocationofthepeak
feature in other transport or thermodynamic properties. in A. Both features have little temperature dependence
This peak is not resolved in measurements at 1.5K25. below 0.5K12. This is seen more clearly in the inset to
Fig. 2.
Precisely at B , there is a sharp kink in the ther-
MMT
mopower. S/T(B) changes slope suddenly, within the The successive approximations involved in estimating
resolution of our measurement, about 0.05T. This fea- the q-ratio impose severe limitations on the validity of
ture is more clearly seen in Fig. 2. This kink is resolved this test, in particular if either N(ε) or τ have a strong
upto the highesttemperature thatwe measure(0.57K). dependence on energy. In a narrow band system the
This temperature dependence is similar to that of the magneticfieldmaythencausestrongvariationsineither
narrowfeatureintheHalleffectatBMMT,whichisclear quantity as the Zeeman energy increases. At a topolog-
and essentially unchanged below 0.5K5. ical transition of the Fermi surface in three dimensions
a. Sign of the thermopower The sign of the ther- either(∂lnN(ε)/∂ε) or(∂lnτ(ε)/∂ε) divergeswhen
εF εF
mopower is often used as an indicator of the sign of the N(ε) does not. Sharp features in S/T without ther-
dominant carriers. An electron band would make a neg- modynamic counterparts are then possible. The peak
ative contribution to the thermopower, and a hole band at 1T may be one such feature. One way of distin-
apositivecontribution. Inthecaseofheavyfermionma- guishing whether it is the scattering or the thermoyd-
terials, this general trend is inverted. Ce-based heavy namic term that is important might be to measure the
fermion materials typically have a positive thermopower Nernsteffect,whichis(insomelimit)onlydependenton
inthe lowtemperaturelimit26,eventhoughthey usually (∂τ(ε)/∂ε) 29.
εF
4
C/T.
0.6 424350mmKK 0.6 Interestingly, a superposition of these two models
106mK ol) wouldbeabletoaccountforboththetransportandther-
mol) 0.5 0.5 2J/Km mhaovdeyanapmeaickainnotmhealdieesn.sitSyucohf satactoems bariniseidngmfroodmelawnoounld-
2J/K 0.4 0.4 45 ( vanishingbandcoincidentwiththeedgeofanotherband.
1/2 ( + 0. ThisideaisillustratedinFigure3a,whereC/T isshown
R)KW 0.3 0.3 e/Tq athriesinavgefrraogme tCw/oTs)u.chInconthtreibsuatmioensve(iinn,thifiswceasceownesttraukcet
A/ 0.6 Av
γ=( 0.2 0.2 SN S(B)/T fromtwocontributions,similarinshapetothose
0.5 γ = - calculatedfor the Lifshitz and peak models, we arriveat
B a curve that bears a qualitative resemblence to the one
0.1 MMT 0.1
7 8 observed(seeFig.3c,whichshowsaweightedsumofthe
0 2 4 6 8 10 12 twomodelcalculationstoillustratethis. Aweightedsum
Magnetic Field (T) is appropriate for the thermopower since S = P Siσi).
i σ
While we can envisage a bandstructure that contains
both a peak and a band egde, if it is to be more than
FIG.2: (Coloronline)Comparingthermopowertothermody-
simply serendipitous that they are both traversedat the
namics. We calculate the electronic specific heat coefficient,
γ, in two ways. On the left scale we plot γ obtained from same magnetic field then a further physical connection
the Kadowaki-Woods ratio, by using the A-coefficient of the is required. There must be some reason for the peak to
electrical resistivity in the formula A/γ2 = RKW. On the be pinned to the band edge. We note that similar si-
right scale we use the q-ratio (see text) to scale the ther- multaneous signatures in transport and thermodynamic
mopower data into units of γ. We also invert the sign and properties have been recently observed in another ma-
offset it by 0.45Jmol−1K−2 to match the left scale at zero terial10, making a generic mechanism more likely than
field. An offset is justified by the presence of other bands,
coincidence.
while thesign inversion is appropriatefor heavyfermion ma-
One possibility is that the ‘peak’ contribution arises
terials. Clearly, the structure in the thermopower near the
fromfluctuationsthatoriginateinthemetamagneticfluc-
MMT is more complex than the thermodynamic signature.
tuations between states of low and high magnetisation.
TheinsetshowsanenlargementatBMMT showingthesharp
kinkinthethermopower-deriveddatawhichpersiststohigher These mayalsobe viewedastopologicalfluctuations be-
temperatures. tween the state with an extra Fermi surface pocket, and
the one where it has vanished. The Lifshitz transition
is a kind of quantum phase transition, and therefore we
III. RIGID BAND SHIFT MODELS expectaregimeoffluctuationstosurroundit. Whileone
band vanishes, these fluctuations would renormalise the
electronic density of states in every band, most strongly
We consider now two simple model calculations that
at B .
help us to understand the generic features of transport MMT
While there has been some recent interest in quan-
and thermodynamics. Both models use a single, spin-
tumtopologicaltransitionsinmetamagneticsystems8,11,
split spherical band with parameters drawn from dHvA
a full theoretical treatment is still lacking and the ideas
measurements. In the ‘Lifshitz’ model (also discussed in
presented here remain quite speculative. It is clear that
Ref. 5), the splitting is so strong that one spin-subband
non-interacting models with rigid band shifts are insuf-
becomescompletelydepopulatedatB . Inthe‘peak’
MMT
ficient to explain the data we obtain and it may there-
model, ε is driven through a sharp peak in the density
F
fore be interesting to explore field dependent renormal-
of states at B (as suggested in Ref. 17). We ex-
MMT
ized band calculations, as have been performed success-
tend the transport calculation of Ref. 5 to include the
fully for YbRh Si 30, where the field dependence of the
thermopowerby applying the Mott formula,andwe also 2 2
heavy fermion bandstructure is accounted for. As an
calculate the electronic specific heat. These calculations
illustration of the possible effect of strongly field depen-
areshowninFigure3. Aswecansee,C/T canbewellde-
dent parameters, we show again in Figure 4 the Lifshitz
scribed by the peak model. However,there are no sharp
modelcalculation,repeatedwith aZeemanenergytaken
features in the transport propeties. The Lifshitz model,
to be proportional to the experimental magnetisation.
meanwhile, can reproduce well the sharp features seen
Thetransitionhasthesamegeneralfeatures,butiscom-
in the Hall effect. The calculated S(B)/T also has an
pressed into a narrower field range.
asymmetric shape and a sharp kink, qualitatively simi-
lar to the data. It is difficult to otherwise reproduce the
sharpstructure seen in the transportsimply by invoking
IV. SUMMARY
sharp features in N(ε); if this were the case one would
expecttoseefeaturesofsimilarwidthinmagneticfieldin
boththermodynamicandtransportproperties. However, We have presented high resolution measurements of
the Lifshitz model cannot reasonably produce a peak in the thermal transport properties of CeRu Si across the
2 2
5
0.8 MMT. The thermopower, like the Hall effect, shows a
2Kmol) 0.7 a) BMMT swhiathrpoufteaatusirnegautlatrhceonMtrMibTutwiohnicthoitshedidffiecnuslittytoofesxtpaltaeisn.
J/
T ( 0.6 However,thethermodynamicpropertiesattheMMTare
C/ not divergent. A topologicaltransition of the Fermi sur-
at 0.5 Lifshitz model
He Peak model face is one way in which transport anomalies can be ex-
c. 0.4 weighted sum plained. Wehavetriedtoshowhowthethermodynamics
e
Sp might also be accounted for if quantum fluctuations are
considered,butweawaitarigoroustheoreticaltreatment
m) 0.5 b)
c of the interacting Lifshitz point.
µΩ 0.4 ρ
ρy ( 0.3 xx ρxy
vit
sti 0.2
si
Re 0.1
2) 0
V/K -0.8 c)
µ
T (
S/ -1.2
er
w
o -1.6
p
o
m
er -2
h
T
0 2 4 6 8 10 12
Magnetic Field (T)
FIG. 3: (Color online) Model calculations of transport and
thermodynamic properties at zero temperature. In the
‘Lifshitz’ model (dark red), the spherical Fermi surface is
spin-split until one band becomes completely unoccupied at
BMMT. Thetotalnumberofoccupiedstatesisconserved. In
the‘peak’model(lightblue),theFermienergypassesthrough
a peak in the density of states at BMMT. A combination of
these models (see text for details, dashed green) can qualita-
tively reproduce key features of both transport and thermo-
dynamicproperties. a)Thespecificheat,C/T. Theobserved
specific heat can be well reproduced by the peak model. b)
TheresistivityandHalleffectonlyhavesharpfeaturesinthe
Lifshitz model calculation. c) The thermopower of the Lif-
shitzmodelhasanasymmetricstructureandasharpkinkat We acknowledgehelpful discussionswith V. Zlati´c,M.
BMMT. Vojta and C. Geibel.
1 I. Lifshitz, Soviet Physics JETP 11, 1130 (1960). Matter 22 (2010).
2 Y. Blanter, M. Kaganov, A. Pantsulaya, A. Varlamov, 9 L.Malone,T.D.Matusda,A.Antunes,G.Knebel,V.Tau-
Phys. Rep.245, 159 (1994). four,D.Aoki,K.Behnia,C.Proust,andJ.Flouquet,Phys.
3 A. Hackl and M. Vojta, Phys. Rev. Lett. 106, 137002 Rev.B 83, 245117 (2011).
(2011). 10 E. A. Yelland, J. M. Barraclough, W. Wang, K. V.
4 M. M. Altarawneh, N. Harrison, S. E. Sebastian, L. Bal- Kamenev,andA.D.Huxley,NaturePhysics7,890(2011).
icas, P. H. Tobash, J. D. Thompson, F. Ronning, and 11 A. W. Rost, S. A. Grigera, J. A. N. Bruin, R. S. Perry,
E. D. Bauer, Phys.Rev.Lett. 106, 146403 (2011). D. Tian, S. Raghu, S. A. Kivelson, and A. P. Mackenzie,
5 R.Daou,C.Bergemann,andS.R.Julian,Phys.Rev.Lett. arXiv 1108.3554v1 (2011).
96, 026401 (2006). 12 F. Weickert, M. Brando, F. Steglich, P. Gegenwart, and
6 L. P. Gorkov and P. D. Grigoriev, Phys. Rev. B 73, M. Garst, Phys. Rev.B 81, 134438 (2010).
060401(R) (2006). 13 A. J. Millis, A. J. Schofield, G. G. Lonzarich, and
7 H.Kotegawa,V.Taufour,D.Aoki,G.Knebel,andJ.Flou- S. A.Grigera, Phys.Rev.Lett. 88, 217204 (2002).
quet,J. Phys.Soc. Jpn 80 083703 (2011). 14 R.Daou,C.Bergemann,andS.R.Julian,PhysicaB378-
8 M. Imada,T. Misawa, andY.Yamaji, J. Phys.: Condens. 80, 807 (2006).
6
15 M.Takashita,H.Aoki,T.Terashima,S.Uji,K.Maezawa,
0.02 a) R.Settai,andY.Onuki,J.Phys.Soc.Jpn65,515(1996).
ev) 16 J. Flouquet, P. Haen, S. Raymond, D. Aoki, and
y E ( 0.01 G. Knebel, Physica B 319, 251 (2002).
Energ 0 EE ∝∝ MH 17 YR..ASeotktia,iT,.YD..OMnautksiu,dEa,.HY.aSmuagmawotaor,a,YH..HSaagtoa,,HA..OVh.kuAnni-,
eman -0.01 18 dKr.eKeva,daonwdaoktihaenrds,SJ..WMoagond.s,MSaoglind.SMtaatteerC.1om77m,u2n71.5(189,9580)7.
e
Z -0.02 (1986).
ρ 19 S. Kambe, J. Flouquet, P. Haen, and P. Lejay, J. Low
cm) 0.5 b) xx -0.8 2µV/K) 20 TDe.mApo.kiP,hCy.sP.a1u0l2se,n4,7T7.(D19.9M6)a.tsuda,L.Malone,G.Knebel,
µΩ 0.4 T ( P.Haen,P.Lejay,R.Settai,Y.Onuki,andJ.Flouquet,J.
ρy ( 0.3 ρxy -1 er S/ Phys. Soc. Jpn 80 053702 (2011).
Resistivit 00..12 -1.2 ermopow 2221 (MK2.0.0MS2a)ity.oa,keY.anKdoikOe.,NS.arKikaityaon,oJ,.NP.hMyse.toSkoic,.HJ.pKna7d1o,wa8k6i7,
BMMT -1.4 Th and S. Kawarazaki, J. Phys.Soc. Jpn 73, 3418 (2004).
00 2 4 6 8 10 12 23 A. Amato, D. Jaccard, J. Sierro, P. Haen, P. Lejay, and
Magnetic Field (T) J. Flouquet, J. Low Temp. Phys.77, 195 (1989).
24 M.Sera,S.Kobayashi,M.Hiroi,N.Kobayashi,H.Ohkuni,
and Y. Onuki,Phys.Rev.B, 56, 13689 (1997).
FIG.4: (Color online) Lifshitzmodelcalculation with exper- 25 R.Bel, Ph.D. thesis, Universit´eParis 7 (2004).
imental field dependence. The Zeeman energy, E, is plotted 26 K. Behnia, D. Jaccard, and J. Flouquet, J. Phys.: Con-
in the top panel. Dashed lines show the results for E ∝ B dens. Matter 16, 5187 (2004).
and solid lines for E ∝ M. The peak in the measured ther- 27 K.MiyakeandH.Kohno,J.Phys.Soc.Jpn74,254(2005).
mopower (dark red, right axis) is asymmetric with respect 28 V. Zlati´c, R. Monnier, J. K. Freericks, and K. W. Becker,
to the critical field BMMT, with a greater apparent weight Phys. Rev.B 76, 085122 (2007).
for B < BMMT. If the Zeeman splitting is instead taken 29 K. Behnia, J. Phys.: Condens. Matter 21, 113101 (2009).
to be proportional to the experimental magnetisation (from 30 G. Zwicknagel, J. Phys.: Condens. Matter 23, 094215
Ref. 16, dashed lines) the features in the calculation appear (2011).
much narrower.
