Metamagnetic quantum criticality in Sr3Ru2O7 studied by thermal expansion

0.24 MB·

by  P. Gegenwart

APS/123-QED
Metamagnetic quantum criticality in Sr Ru O studied by thermal expansion
3 2 7
P. Gegenwart(1), F. Weickert(1), M. Garst(2), R.S. Perry(3,4,5), Y. Maeno(4,5)
(1) Max-Planck Institute for Chemical Physics of Solids, D-01187 Dresden, Germany
(2)Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA
(3)School of Physics and Astronomy, University of St. Andrews, Fife KY16 9SS, Scotland
(4)International Innovation Center, Kyoto University, Japan
(5)Department of Physics, Kyoto University, Japan
6
(Dated: February 2, 2008)
0
0 Wereport low-temperaturethermalexpansionmeasurements on thebilayerruthenateSr Ru O
3 2 7
2 as a function of magnetic field applied perpendicular to the Ruthenium-oxide planes. The field-
n dependence of the c-axis expansion coefficient indicates the accumulation of entropy close to 8
a Tesla, related to an underlying quantum critical point. The latter is masked by two first-order
J metamagnetic transitions which bound a regime of enhanced entropy. Outside this region the
1 singularthermalexpansionbehavioriscompatiblewiththepredictionsoftheitineranttheoryfora
3 two-dimensional metamagnetic quantum critical end point.

] PACSnumbers: 71.10.HF,71.27.+a
l
e
-
r Quantum phase transitions (QPTs) in itinerant elec- deedbe tunedby changingthe angleofthe appliedmag-
t
s tron systems are of extensive current interest in con- netic field from 1.25 K for fields perpendicular to the
.
t densed matter physics because they are not only at the c-axis towards zero for H k c [7]. For this field orienta-
a
m origin of unusual finite temperature properties but also tion,non-FermiliquidbehavioroccursrelatedtoaQCEP
promotetheformationofnewstatesofmatterlikeuncon- close to 8 T [5]. However, the electrical resistivity at
-
d ventionalsuperconductivityinheavyfermionsystems[1]. fieldsveryclosetothe metamagneticregionhasrevealed
n Dilatometricstudiesareespeciallysuitabletoinvestigate additional features: In a new generation of high-quality
o quantum criticality since they directly probe the sensi- Sr Ru O singlecrystalswitharesidualresistivityaslow
3 2 7
c
tivity of thermodynamics resulting from the QPT being as0.4µΩcm[8]threepeaksinthesusceptibilityandmag-
[
susceptible to pressure-tuning. In particular, near any netostrictionhavebeenobservedbelow1K.Thefirstone

3
pressure-tuned quantum critical point (QCP) the ther- at 7.5 T marks a cross-over,whereas the two latter ones
v
malexpansion,whichat constantpressurequantifies the at 7.85 and 8.07 T are first-order metamagnetic transi-
9
temperature dependence of the sample volume, is more tions. Theyconfine alow-temperature(T ≤1K)regime
5
3 singularthanthespecific heat[2,3]. Below,weusether- in which the residual electrical resistivity is strongly en-
7 mal expansion to study a certain type of QCP associ- hanced indicative of strong elastic scattering and which
0 ated with metamagnetism in metals. Generally, metam- disappearsby rotatingthe fieldformorethan10degrees
5
agnetism describes the sudden rise of the magnetization away from H k c [9]. It has been proposed [9] that this
0
M(H) as a function of the applied magnetic field H. A behaviormightresultfromtheformationofasymmetry-
/
t
a line of first-order metamagnetic transitions in the mag- broken phase characterized by a spin-dependent Pomer-
m netic field–temperature plane has an end point (H∗,T∗) anchuk deformation of the Fermi surface. Alternatively,
- beyondwhichthetransitionbecomesacontinuouscross- a scenarioofphase separationleading to chargeinhomo-
d over. Anespeciallyinterestingsituationnowariseswhen geneities within the bounded regime has been discussed
n ∗
the temperature T can be tuned to zero giving rise to [10].
o
an isolated quantum critical end point (QCEP) [4]. In this Letter, we report a c-axis thermal expansion
c
: The material that initiated the interest in QCEPs is studyonSr Ru O attemperaturesdownto100mKand
v 3 2 7
the ruthenate Sr Ru O [5], which crystallizes in the bi- mT field steps close to the metamagnetic regime for H k
i 3 2 7
X layered version of the perovskite structure. Its meta- c. Weobservequantumcriticalbehaviorcompatiblewith
r magnetic transition conforms to the picture of a field- thepredictionsforatwo-dimensional(2D)metamagnetic
a
induced Stoner transition given the absence of a dra- QCEP [4]. Distinct anomalies in thermal expansion and
matic change of the Fermi surface [6] and its highly en- electricalresistivityattheboundedregimearediscussed.
hanced Sommerfeld-Wilson ratio. Sr Ru O shows an The thermal expansion measurements were performed
3 2 7
almost isotropic zero-field susceptibility at low temper- onahigh-qualitysinglecrystalwith1.7mmlengthalong
atures, and metamagnetism occurs both between 5 and the c-axis, grown by floating zone technique [8]. Length
6 T for fields in the Ruthenium-oxide planes as well as changes along the c-axis have been detected by utiliz-
near 7.8 T for H kc [5]. A detailed study onsingle crys- ing a high-resolutioncapacitivedilatometer in a dilution
tals with a residual resistivity of about 3 µΩcm has re- refrigerator. Below, we use nominal values for the field
∗
vealedthatthetemperature,T ,ofthecriticalendpoint applied with aid of a 20 T superconducting magnet. For
of a line of first-order metamagnetic transitions can in- comparison with corresponding susceptibility measure-
2
10 T
Sr Ru O α T∼|h|d+21
H 3// c2 7 III
II II T∼|h|
5 m H (T) αmax∼|h|3(dd−+51) I QCEPI
0 0 h
-1K) 6
-610 77.5 I ∼|h|−31Td−32
(c 7.8
a 0 8.2 II ∼ |h|T−d+34
8.5 III
10 ∼|h|d−36T
-5 |h| |h|d+21 T
0 1 2 3 4 5
T (K)
FIG. 2: Behaviour of the thermal expansion, α, in d dimen-
sions near the metamagnetic QCEP of (1), as a function of
FIG. 1: Linear c-axis thermal expansion coefficient of temperature,T,anddistancetothecriticalfield,h∝H−H .
c
Sr Ru O as α vs T at various magnetic fields H k c in-
3 2 7 c Asexplainedinthetext,α(−h)=−α(h). Notethattheinter-
dicated bydifferent symbols.
mediateregimeIIshrinkstozerowithh. Ind=2additional
logarithmiccorrectionsarepresent;II:α∼|h|−1/3log T ,III:
|h|
α∼h/(T2log 1).
ments [9] they need to be offset by 0.037 T. The lin- T
ear thermal expansion coefficient α =d(∆L /L )/dT is
c c c
obtained by calculating the slope of the relative c-axis
length change in temperature intervals of 40 mK. Ad- The leading contributions to the thermal expansion,
ditionally, the electrical resistivity of a small plate-like α = Vm−1(dV/dT)p,H = −Vm−1(dS/dp)T,H (Vm: mo-
piece from the same batch has been measured using a lar volume) will generically derive from the pressure
low-frequency four-point ac-method. The residual resis- dependence of the most relevant coupling, i.e. h or,
tivity ratio of this sample equals 150. equivalently, the critical magnetic field, Hc = Hc(p).
Fig. 1 shows the temperature dependence of α in dif- Derivatives of the free energy with respect to H and p
c
ferent fields between 6 and 10 T for temperatures up then probe the same thermodynamic information,which
to 4 K. Strikingly, the thermal expansion changes sign would naturally explain the close correspondence be-
across the metamagnetic cross-over. This was also ob- tween magnetostriction and susceptibility observed in
served near the metamagnetic cross-over in the heavy- Sr3Ru2O7 [9]. For the critical part of the thermal ex-
fermioncompound CeRu2Si2 [11]. Moreover,as the crit- pansion we get the relation α=ΩVm−1(∂S/∂H)T, where
icalfieldHc isapproachedthepronouncedpeakinαc in- Ω ≡ (dHc/dp)H=Hc ≈ 5.6 T/Gpa [12] is a measure for
creases, narrows and shifts towards lower temperatures. the (linear) hydrostatic pressure dependence of Hc. The
Thecharacteristicshapeofthethermalexpansionasa corresponding relation also holds for the c-axis thermal
function of T and H can be understood within the sce- expansionandthec-axisuniaxialpressuredependenceof
narioofaQCEP.AsoutlinedinRef.[4],nearthecritical the critical field, which will explain the observed sign
magnetic field, Hc, thermodynamics is governed by the change of αc across the metamagnetic transition, see
longitudinal fluctuations of the magnetic polarization of Fig. 1. Indeed, as detailed in [13] a change of sign of the
theFermiseadescribedbyanIsingfieldψwiththeaction thermal expansion is a generic phenomenon near QCPs.
The soft fluctuations associated with the T = 0 criti-
S[ψ]= ddk 1 1 r +k2+ |ωn| |ψ(ω ,k)|2 cal point enhance the entropy at finite T; the entropy
Z (2π)dβ 2(cid:20) 0 k (cid:21) n will therefore increase when H is approached, which is
Xωn c
reflected in a sign change of α ∝ (∂S/∂H) across the
β u T
+ dτ ddx ψ4(τ,x)−hψ(τ,x) . (1) transition. In particular, the accumulation point of en-
Z0 Z h4! i tropy is identified by a vanishing thermal expansion.
When the system is tuned to its QCEP, r ≈ 0, which Within the theory (1) this accumulation of entropy
0
will be assumed in the following, these fluctuations are is expected exactly at H = H ensured by the (emer-
c
very soft near the metamagnetic transition resulting in gent) reflection symmetry ψ → −ψ at h = 0, leading
strong thermodynamic signatures. The deviation of the to the property α(−h) = −α(h) near the QCEP. The
magnetic field from its critical value, h ∝ H −H , then behaviour of the thermal expansion deriving from (1) is
c
acts as the control parameter of metamagnetism. summarized in Fig. 2. If the system is tuned away from
3

10-5 10-5
30
Sr Ru O
-6-2T/ (10 K)c 12000 ~ |H- Hc|-4/3 3 T (K2 )01..722 --4/3624/3HH-)| (10KT)c 10-6 m0H (56667T..)58 m0H (888T...)01275 10-6 --4/3624/3HH-)| (10KT)c
a a ~ |H- H|-4/3 (0 7.2 8.35 (0
-10 c Tm/ | 10-7 777...3445 889..58 10-7 Tm/ |
3 b a 7.5 9.5 -a
a > 0 a < 0 7.6 10
7.65
7.7 a b
2 10-8 10-8
K) a ~ |H- H|2/3 10-310-210-1100101102103 10-2 10-1 100 101 102 103
T ( a m =ax 0 c T 3|m (H- H)|-7/3 (K3T-7/3)
1 a 0 c
min
l
max FIG. 4: (color online) Linear c-axis thermal expansion co-
d2r /dT2
efficient of Sr Ru O rescaled as α /T ×|µ (H−H )|4/3 vs
0 3 2 7 c 0 c
5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 T3×|µ (H−H )|−7/3 withµ H =7.845T(onadouble-log
0 c 0 c
m H (T) scale)forlow-field(a)aswellashigh-field(b)data. For7.4T
0 ≤µ H ≤8.07Tdatafortemperaturesbelow1.1Khavebeen
0
omitted. Lines indicate α ∝|µ (H−H )|T−2.
c 0 c
FIG. 3: (color online) (a): Field dependence of the ther-
mal expansion coefficient of Sr Ru O as α /T vs µ H at
3 2 7 c 0
0.2 K (open circles) and 1.2 K (closed squares). The green
solidlinesinFig.3a). Two-dimensionalspinfluctuations
open circles indicate data in the non-Fermi liquid regime,
∂2α /∂T2 6= 0. The red solid lines show a fit |H −H |−4/3 were also observedin inelastic neutron scattering [14] at
c c
with µ H = 7.845 T. (b): Corresponding (T,H) diagram. zeromagneticfieldwhichrevealedthattheyareconfined
0 c
Triangles and crosses show positions of thermal expansion to the RuO2 bilayers of the crystal structure. Close to
extrema and zeros, respectively. Squares and diamonds in- the metamagnetic region, the low-T thermal expansion
dicatepositionsofpeaksinmagnetostrictionandind2ρ/dT2, in Fig. 3a shows complicated behavior with three sign
respectively, see also Fig. 5. changesinanarrowfieldinterval,whichareattributedto
thefinestructurenearH mentionedintheintroduction.
c
Outside this field region, the zeros of α are independent
the metamagnetic transition, h 6= 0, the field ψ attains of H, see Fig. 3b. The positions of thermal expansion
a non-zero expectation value. At low temperature its extrema follow a |H−H |2/3 dependence as expected in
c
response is non-linear, ψ ∝ |h|1/δ, with the mean-field d = 2 (red line in Fig. 3b). However, at lowest temper-
exponent δ = 3. Thermodynamics is determined by the atures they do not coincide at a single (quantum critical
fluctuations around this mean-field solution giving rise end) point at T = 0 but merge with the two lines of
to regimes I and II. The cross-over I/II is characterized first-order transitions. The presence of these first-order
by incipient deviation from Fermi-liquid behaviour. At transitionsmightalsobeattheoriginofthemissingsym-
elevated temperatures however thermal fluctuations will metry α(−h)=−α(h) already apparent in Fig. 1.
stabilize the mean-field potential resulting in a weaker, The scaling-plotin Fig.4analyzesthe full shape ofα
c
linearresponse,ψ ∝h. Inthis linearresponseregime III in more detail; however, omitting data at temperatures
the behaviour of α is captured by renormalized mean- T ≤ 1.1 K for fields very close to H . The thermal ex-
c
field theory giving a maximum of |α| at the crossover pansion has been rescaled such that a data collapse for
II/III followed by a steep decline at higher T. the low-andhigh temperature asymptotes,regime I and
Forcomparisonwithexperimentthemagneticfieldde- III,shouldoccurfora2DQCEP.Note thatthe presence
pendence of the thermal expansion coefficient, α /T, is oftheintermediateregimeIIpreventsafullcollapseonto
c
shown in Fig. 3a. At elevated temperatures (1.2 K) the a single scaling curve and deviations in this regime are
behaviorresemblesthatobservednearthemetamagnetic even theoretically expected. The scaling behaviour of
cross-over in CeRu Si [11] with a sign change of α /T the raw data is remarkable given the uncertainties asso-
2 2 c
occuring at µ H = (7.845±0.005) T. Using this value ciated with a possible small linear in T background (see
0 c
forthecriticalfield,wecandescribethefielddependence µ H =7.8TinFig.1). Moreover,inthelow-fieldregion,
0
inside the Fermi liquid regime by α /T ∝ |H − H |−ǫ H <H ,theadditionalanomalyat7.5Tleadstodistor-
c c c
with ǫ = 1.35±0.1, in agreement with the expectation tions atlow temperatures. The lines indicate the high-T
for two-dimensional (2D) spin fluctuations (ǫ = 4/3 red asymptotes expected in d=2. Whereas in the high-field
4

first entered and then left upon cooling. This leads to
m H (T) distinct anomalies in both the thermal expansion and
10 T (K)1.0 0 7.8 the electrical resistivity observed, e.g., at a field µ0H =
7.82 8.07Tthatareabsentatlargerfieldswhenthenewstate
0.5 7.84
1) 5 7.87 is not entered. The shape of the bounded region has in-
-K
-60 0.0 7.8 7.9 8.0 m 8H.1 (T) 7.9 teresting consequences for its entropic properties. From
(1c 0 88..0175 the Clausius-Clapeyronrelationfollows that the entropy
a 0 increaseswhenthisstateisentereduponcrossingthefirst
ordertransitionsatconstanttemperature. Apossibleex-
a
planationfortheenhancedentropymightbethatthesoft
-5
fluctuationsarenotcompletelycutoffatthecrossoverbut
b m 0H (T) rathersurvivewithintheboundedregime. Notethatthe
3 7.52
K) 7.77 formation of the bounded state at T ≈ 1 K leads only
m/ 7.82 to a small dip in the thermal expansion (Fig. 5a), and
c 2 7.92
a pronouncedpeak characteristicfor the presence of soft
8.02
mWT ( 8.07 fluctuations alsodevelops within the bounded regionbe-
d
rd/ 1 8.5 low 1 K at a magnetic field of e.g. 7.84 T. The observa-
tion of an enhanced entropy is also consistent with the
0 NMR results [15] which showed that quantum critical
0.0 0.5 1.0 1.5 2.0 2.5 3.0
fluctuations near the metamagnetic field persist within
T (K)
the bounded region down to lowest temperatures.
FIG. 5: (color online) (a): Temperature dependence of the These results place additional constraints on the pos-
thermalexpansioncoefficientαc (a)andresistivityderivative siblenatureoftheboundedregion. Itsenhancedentropy
dρ/dT (b) of Sr Ru O at various magnetic fields H k c.
3 2 7 as well as the absence of phase transition signatures in
Arrows indicate anomalies associated with theentryinto the
thethermalexpansionatthecrossoversnear1Karedif-
bounded state. The inset displays the (T,H) phase diagram
ficult to reconcile with a symmetry-breaking mechanism
(cf. Fig. 3b) for the region very close to the metamagnetic
transitions. Red circles, and green diamonds mark positions for the phase formation as proposed in [9] but it cannot
of arrows in (a) and (b), respectively. be excluded. If the incorporation of fluctuation effects
in a phase separation scenario [10] is able to explain the
experimentalsignaturesase.g.the shapeofthe bounded
regionthefitisnotcompulsory,thebehaviourinthelow-
region remains to be seen. Thus, further theoretical and
fieldregionisconvincinglydescribedbytwo-dimensional
experimental work is needed for clarification.
metamagnetic fluctuations. Note however that a recent
NMR study [15] indicates that the full fluctuation spec- Helpful discussions with S.A. Grigera, A.P. Macken-
trumnearthecriticalfieldmightalsocontaininaddition zie, A. RoschandP.W¨olfle aregratefully acknowledged.
an antiferromagnetic component. P.G.thankstheDeutscheForschungsgemeinschaft(DFG
Atlast,weconcentrateonthefieldregimenear8Tbe- grantGE 1640/1-1)and RoyalSociety (UK) for support
tween the two first-order transitions [8] (squares in inset of his stay at the St. Andrews University, M.G. is sup-
ofFig.5a)whereatlowtemperaturesanenhancedresid- ported by DFG grant GA 1072/1-1.
ual resistivity was observed [9]. As shown in Fig. 5 this
regime in the (T,H) diagramis bounded in temperature
by a cross-over line connecting the two first-order tran-
sitions where the thermal expansion and especially the
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