Table Of ContentPhysical Review Letters 92, 047002 (2004)
Gap Structure of the Spin-Triplet Superconductor Sr RuO
2 4
Determined from the Field-Orientation Dependence of Specific Heat
K. Deguchi,1,∗ Z. Q. Mao,1,† H. Yaguchi,1 and Y. Maeno1,2
1Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
4 2Kyoto University International Innovation Center, Kyoto 606-8501, Japan
0 (Dated: February 2, 2008)
0
2 Wereportthefield-orientationdependentspecificheatofthespin-tripletsuperconductorSr2RuO4
underthemagneticfieldalignedparalleltotheRuO2planeswithhighaccuracy. Belowabout0.3K,
n
striking4-foldoscillations ofthedensityofstatesreflectingthesuperconductinggap structurehave
a
been resolved for the first time. Wealso obtained strong evidence of multi-band superconductivity
J
andconcludedthatthesuperconductinggapintheactiveband,responsibleforthesuperconducting
0
instability, is modulated with a minimum along the[100] direction.
3
PACSnumbers: 74.70.Pq,74.25.Bt,74.25.Op,74.25.Dw
]
n
o
c Sincethediscoveryofitssuperconductivity[1],thelay- ture inthe active bandis currentlyof prime importance.
- ered ruthenate Sr RuO has attracted a keen interest The field-orientation dependent specific heat is a direct
r 2 4
p in the physics community [2]. The superconductivity of measure of the QP density of states (DOS) and thus a
u Sr RuO has pronounced unconventional features such powerfulprobeoftheSCgapstructure[25,26,27,28]. In
2 4
s
as: the invariance of the spin susceptibility across its this Letter, we report high precision experiments of the
.
at superconducting (SC) transition temperature Tc [3, 4], specific heatas a function of the angle betweenthe crys-
m appearance of spontaneous internal field [5], evidence tallographic axes and the magnetic field H within the
for two-component order parameter [6] and absence of a RuO plane. We reveal that the SC state of Sr RuO
- 2 2 4
d Hebel-Slichterpeak[7]. Thesefeaturesarecoherentlyun- has a band-dependent gap and that the gap of its active
n
derstood in terms of spin-triplet superconductivity with SCbandhasstrongin-planeanisotropywithaminimum
o
the vector order parameter d(k) = zˆ∆ (k +ik ), rep- along the [100] direction, as illustrated in Fig. 1.
c 0 x y
[ resenting the spin state Sz = 0 and the orbital wave Single crystals of Sr RuO were grown by a floating-
2 4
3 function with Lz =+1, called a chiral p-wave state. zone method in an infrared image furnace [29]. After
v The above vector order parameter leads to the gap specific-heatmeasurementsontwocrystalstoconfirmthe
6 ∆(k) = ∆0(kx2 + ky2)1/2, which is isotropic because of reproducibilityof salientcharacteristicssuchasa double
6 the quasi-two dimensionality of the Fermi surface con- SC transition [30], the sample with T = 1.48 K, close
c
3 sisting of three cylindrical sheets [8]. However, a num- to the estimated value for an impurity and defect free
1
ber of experimental results [9, 10, 11, 12, 13] revealed specimen (T = 1.50 K) [31], was chosen for detailed
1 c0
3 the power-law temperature dependence of quasiparticle study. This crystal was cut and cleaved from the sin-
0 (QP) excitations, which suggest lines of nodes or node-
/ likestructuresintheSCgap. Therehavebeenmanythe-
t
a oreticalattempts(anisotropicp-waveorf-wavestates)to
m
resolvethiscontroversy[14,15,16,17,18,19]. Although
- all these models suggest a substantial gap anisotropy,
d
magnetothermalconductivitymeasurementswiththeap-
n
o plied field rotated within the RuO2 plane down to 0.35
c K revealed little anisotropy [20, 21]. To explain those
: experimental facts as well as the mechanism of the spin-
v
i tripletsuperconductivity,severaltheories[22,23],taking
X
the orbital dependent superconductivity (ODS) into ac-
r count [24], have been proposed. In these models, there
a
are active and passive bands to the superconductivity:
the SC instability originatesfromthe activebandwith a
large gap amplitude; pair hopping across active to pas-
sive bands leads to a small gap in the passive bands.
FIG.1: Left: Electronicspecificheatdividedbytemperature
The gap structure with horizontal lines of nodes [22] or
Ce/T for H k [100], as a function of field strength and tem-
strong in-plane anisotropy [23] in the passive bands was
perature. AcontourplotisshownonthebottomH−T plane,
proposed. with the same color scale as the 3D plot. Right: Supercon-
In order to identify the mechanism of the spin-triplet ductinggapstructurefortheactivebandγ deducedfromthe
superconductivity, the determination of the gap struc- presentstudy,correspondingtod(k)=zˆ∆0(sinakx+isinaky).
2
gle crystalline rod, to a size of 2.8 × 4.8 mm2 in the
ab-plane and 0.50 mm along the c-axis. The side of the
crystal was intentionally misaligned from the [110] axis
by 16◦. The field-orientation dependence of the specific
heat was measured by a relaxation method with a dilu-
tionrefrigerator. Sinceaslightfieldmisalignmentcauses
2-fold anisotropy of the specific heat due to the large
H anisotropy (H /H ≈20) [30], the rotation of
c2 c2kab c2kc
the field H within the RuO plane with high accuracy
2
is very important. For this experiment, we built a mea-
surementsystemconsistingoftwoorthogonallyarranged
SC magnets [32] to control the polar angle of the field
H. The twoSCmagnetsareinstalledinadewarseating
on a mechanical rotating stage to control the azimuthal
angle. Withthedilutionrefrigeratorfixed,wecanrotate FIG.2: (a),(b)FieldandtemperaturedependenceofCe/T of
the field H continuously within the RuO plane with a Sr2RuO4 in magnetic fieldsparallel to the[100] (open circle)
2
and [110] (closed circle) directions.
misalignment no greater than 0.01◦ from the plane.
TheelectronicspecificheatC underthein-planemag-
e ponent in the raw data guarantees that the in-plane
netic fields was obtainedafter subtractionof the phonon
field alignment is accurate during the azimuthal-angle
contributionwithaDebyetemperatureof410K.Theleft
rotation. Thus C (T,H,φ) can be decomposed into
panel of Fig. 1 shows C /T for the [100] field direction, e
e φ-independent and 4-fold oscillatory terms, where the
asafunctionoffieldandtemperature. Thefigureiscon-
in-plane azimuthal field angle φ is defined from the
structed fromdata involving13 temperature-sweeps and
[100] direction: C (T,H,φ) = C (T,H)+ C (T,H,φ).
11field-sweeps. Atlowtemperaturesinzerofield,power- e 0 4
C (T,H,φ)/C isthenormalizedangularvariationterm,
law temperature dependence of C /T ∝T was observed, 4 N
e
corresponding to the QPs excited from the line nodes or
node-like structure in the gap.
Now we focus on the field dependence of C /T at low
e
temperature shown in Fig. 1 and Fig. 2 (a). C /T in-
e
creases sharply up to about 0.15 T and then gradually
for higher fields. This unusual shoulder is naturally ex-
plained by the presence of two kinds of gaps [9]. On the
basisofthedifferentorbitalcharactersofthethreeFermi
surfaces (α,β, and γ) [8], the gap amplitudes ∆ and
αβ
∆ are expected to be significantly different [24]. The
γ
normalized DOS of those bands are Nαβ = 0.43 and
Ntotal
Nγ = 0.57 [2]. Since the position of the shoulder in
Ntotal
C /T corresponds well with the partial DOS of the α
e
and β bands, we conclude that the active band which
has a robust SC gap in fields is the γ-band, mainly de-
rived from the in-plane d orbital of Ru 4d electrons.
xy
Figures 2 (a) and (b) show the field and temperature
dependence of C /T under the in-plane magnetic fields
e
H k [100] and H k [110] and indicate the existence of a
slight in-plane anisotropy.
Inthemixedstate,theQPenergyspectrumisaffected
by the Doppler shift δω = ~k·v , where v is the su-
s s
perfluid velocity around the vortices and ~k is the QP
momentum. This energy shift gives rise to a finite DOS
at the Fermi level in the case of δω ≥ ∆(k) [34]. Since
v ⊥ H, δω = 0 for k k H. Thus the generation of
s
nodal QPs is suppressed for H k nodal directions and FIG.3: Thein-planefield-orientation dependenceof normal-
yields minima in Ce/T [25, 26, 27]. ized4-foldcomponentofthespecificheatatseveralfieldsand
Figure 3 shows the field-orientationdependence of the temperatures. Only1.45 Tdata is reduced to1/3. Thesolid
specific heat. The absence of a 2-fold oscillatory com- lines are fits with f4(φ) given in thetext.
3
where C is the electronic specific heat in the normal matethecontributionof∆C /γ T fromtheactiveband
N e N c
state: C = γ T with γ = 37.8 mJ/K2mol. There for the gap structures #1 and #2: ∆C /γ T = (1.22
N N N e N c
is no discernible angular variation in the normal state to 1.07)× 0.57 = 0.70 to 0.61 with the gap minimum
(µ H = 1.7 T > µ H ); possibilities of angular varia- (∆ /∆ = 1/2 to 1/4) [16], while ∆C /γ T =
0 0 c2 min max e N c
tionoriginatingfromexperimentalsetuporotherextrin- 0.75×0.57 = 0.42 with the line nodes [17]. From the
sic contributions are excluded. experimental result ∆C /γ T = 0.73 in Fig. 1 and the
e N c
For fields near H (1.2 T ≤ µ H ≤ 1.45 T), a si- estimatedadditionalcontributionfromthepassivebands
c2 0
nusoidal 4-fold angular variation is observed : C4(φ) ∝ ∆Ce/γNTc ∼ 0.04 [23], d(k) = zˆ∆0(sinakx + isinaky)
f (φ) = −cos4φ. This is consistent with the in-plane si- with the gap minimum is promising for the active band.
4
nusoidalanisotropyofH withthemaximuminthe[110] To facilitate a comparison with theories, although
c2
direction [21, 33]: C = Hc2k[110]−Hc2k[100]dCeHcos4φ. they are presently available only for line-node gaps [26,
4 Hc2k[110]+Hc2k[100] dH 27], we decomposed C into two parts: C (T,0) due
Since H decreases with increasing T, the oscillation e 0
c2 to the thermally excited QPs and ∆C (T,H) due to
amplitude at 1.3 T increases strongly at 0.51 K. For 0
the field induced QPs, consisting of an isotropic com-
µ H < 1.2 T, however, a non-sinusoidal 4-fold angular
0 ponent and a 4-fold anisotropic component A (T,H):
variation approximated as C (φ) ∝ f (φ) = 2|sin2φ|−1 4
4 4 C (T,H,φ)=C (T,0)+∆C (T,H)[1+A (T,H)f (φ)],
is observed. Importantly, a phase inversion in C (φ) oc- e 0 0 4 4
4 A (T,H)f (φ) = C4(T,H,φ), where f was defined pre-
curs across about µ0H = 1.2 T: C4(φ) takes minima at 4 4 ∆C0(T,H) 4
φ= πn(φ= π+πn,n: integer)forµ H <1.2T(µ H ≥ viously for low- and high-field ranges. Figures 4 (a)
2 4 2 0 0
1.2 T), and thus the angular variation for µ H < 1.2 T and (b) show the field and temperature dependence of
0
cannot be due to the in-plane Hc2 anisotropy. Therefore A4. The field dependence of A4 with a maximum of 4%
we conclude that the non-sinusoidal 4-fold oscillations anisotropy at 0.31 K shows a monotone decrease from
originate from the SC gap structure. This result does the delocalized-QP dominant region at low fields to the
not contradictthe previous measurementsof the magne- Hc2-anisotropydominantregionathighfields. The tem-
tothermal conductivity down to 0.35 K [20, 21], which perature dependence of A4 with 3% anisotropy at 0.9
reported little in-plane anisotropy, because these clear T showsasmoothdecreasewithincreasingtemperature.
oscillations emerge only at lower T (T/T ≤0.2). Theseresultsareinsemi-quantitativeagreementwithre-
c
cent theories [26, 27] which predict 4 to 1.5% anisotropy
For the field range 0.15 T < µ H < 1.2 T, where the
0
from gap structures with vertical line nodes.
QPs in the active band γ are the dominant source of in-
In contrastto the theoretical prediction [27], however,
plane anisotropy,we first deduce the existence of a node
at combined low fields (µ H ≤ 0.15 T) and low tem-
or gap minimum along the [100] direction, because C 0
e
peratures (T ≤ 0.3 K) where QPs on both the active
takes a minimum. In addition, we found that the 4-fold
and passive bands are important for the anisotropy, A
oscillationshaveanon-sinusoidal form, approximatedas 4
rapidlydecreases. ThissteepreductionofA isprimarily
C (φ) ∝ 2|sin2φ|−1, since cusp-like features are clearly 4
4
seen at the minima (φ = πn). Strong k dependence of attributable to the non-zero gap minima ∆min of the ac-
2 z tiveγ band(#1). Infact,atthelowesttemperaturesthe
the gapfunction wouldenhance the QP excitations even
if H is parallel to the nodal direction, so that the cusp- 4-fold oscillations are suppressed below a threshold field
like features would have been strongly suppressed [25].
Most of the proposed gap structures can be classi-
fied into four groups as summarized in Table I. #1
and #2 provide the direction of the gap minima con-
sistent with our observation. To distinguish between
#1 with gap minima and #2 with nodes, we examine
the specific heat jump ∆C /γ T at T in zero field.
e N c c
The jump originates mainly from the active band with
large ∆ because of ∆C /γ T ∝ ∂∆2/∂T| . We esti-
e N c Tc
TABLE I: The classified order parameters with the typical
gap structures for Sr2RuO4.
# d(k) direction of node or ∆min Ref.
132 zˆzˆ∆∆zˆ0∆0((k0sx2kinx−akkykx(y2k)+x(ki+xsini+kayik)kyy)) [1[[011001]00]]tinnnooyddgeeassp [[[111786]]] uF4-aIfGtoel.dd4af:rnoim(sao)tt,rho(epbyfi)tAtFi4niegilndttoahntehdseptoeecsmcifiiplclearhtaoetaruytr.edTadhteaepipennodiFneitngs.cae3r,eowfevhtaihllee-
4 (cid:26) zˆ∆zˆ∆0(0kkxz+(kxik+y)ickoys)c2kz horizontal nodes [19] tHhekli[n10e0s]firnomFigt.he2.dTiffweoremnceethiondCseyibeledtwcoeennsisHtenkt r[e1s1u0l]tsa.nd
4
of about 0.3 T, which should correspond to δω = ∆ .
min
However, even at 0.15 T below the threshold, increasing
temperature leads to an increase in A only up to about
4 ∗ Electronic address: [email protected]
0.25 K (Fig. 4 (b)). This behavior is at least in part ex- † Presentaddress: PhysicsDepartment,TulaneUniversity,
plainedbyincluding contributionsofbothfieldandtem-
2001 Percival Stern,New Orleans, LA 70118, USA.
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theoreticalanalysisis needed to examine this possibility. [2] A.P.MackenzieandY.Maeno,Rev.Mod.Phys.75,657
(2003) and thereferences therein.
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[8] A.P. Mackenzie et al., Phys. Rev.Lett. 76, 3786 (1996);
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4
T decreases steeply with decreasing temperature in the (2001); T. Takimoto, ibid. 62, R14641 (2000).
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(1999).
thegapstructureofthepassivebands,theobservedsteep
[17] M.J. Graf and A.V. Balatsky, Phys. Rev. B 62, 9697
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4 (2000).
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In conclusion, we have for the first time revealed the
[19] Y.Hasegawaetal.,J.Phys.Soc.Jpn.69,336(2000);Y.
in-plane anisotropy in the SC gap of the spin-triplet su- Hasegawa and M. Yakiyama, ibid. 72, 1318 (2003); H.
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057001 (2001); J.F. Annett et al., Phys. Rev. B 66,
along the [100]directionwith little interlayerdispersion.
134514 (2002); S. Koikegami et al., ibid. 67, 134517
This gap structure is in good correspondence with the
(2003).
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We thank N. Kikugawa, K. Machida, T. Nomura, H. [27] P. Miranovi´c et al.,Phys. Rev.B 68, 052501 (2003).
Ikeda, Kosaku Yamada, T. Ohmi, M. Sigrist, K. Ishida, [28] T. Park et al.,Phys. Rev.Lett.90, 177001 (2003).
A.P.Mackenzie,S.A.Grigera,K.Maki,andT.Ishiguro [29] Z.Q. Mao et al., Mat. Res. Bull. 35, 1813 (2000).
[30] K.Deguchiet al.,J.Phys.Soc.Jpn.71,2839 (2002); H.
for stimulating discussions and comments. This work
Yaguchi et al.,Phys. Rev.B 66, 214514 (2002).
wasinpartsupportedbytheGrants-in-AidforScientific
[31] A.P. Mackenzie et al., Phys. Rev. Lett. 80, 161 (1998);
Research from JSPS and MEXT of Japan, by a CREST Z.Q. Mao et al., Phys.Rev.B 60, 610 (1999).
grant from JST, and by the 21COE program “Center [32] Cryomagnetics,inc.,“VectorMagnet”allowingahorizon-
for Diversity and Universality in Physics” from MEXT talfieldofupto5Tandaverticalfieldofupto3T.The
of Japan. K. D. has been supported by JSPS Research details of the apparatus is described in : K. Deguchi, T.
Fellowship for Young Scientists. Ishiguro, and Y. Maeno, cond-mat/0311587.
5
[33] Z.Q. Mao et al.,Phys.Rev. Lett.84, 991 (2000).
[34] G.E. Volovik, JETP Lett. 58, 469 (1993).
