Table Of ContentTypeset with jpsj2.cls <ver.1.2> Letter
Evidence for Point Nodes in the Superconducting Gap Function in the Filled
Skutterudite Heavy-Fermion Compound PrOs Sb : 123Sb-NQR Study under Pressure
4 12
Kouta Katayama, Shinji Kawasaki, Masahide Nishiyama, Hitoshi Sugawara1, Daisuke Kikuchi2, Hideyuki
7
0 Sato2 and Guo-qing Zheng
0
2 Department of Physics, Faculty of Science, Okayama University, Okayama 700-8530, Japan
1Department of Mathematical and Natural Sciences, Faculty of Integrated Arts and Sciences, The University of Tokushima,
n
Tokushima 770-8502, Japan
a
J 2Department of Physics, Graduate School of Science, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan
5 (ReceivedFebruary6,2008)
2
Wereport123Sbnuclearquadrupoleresonance(NQR)measurementsofthefilledskutterudite
heavy-fermion superconductor PrOs Sb under high pressures of 1.91 and 2.34 GPa. The
] 4 12
n temperature dependence of NQR frequency and the spin-lattice relaxation rate 1/T indicate
1
o that the crystal-electric-field splitting ∆ between the ground state Γ singlet and the first
CEF 1
r-c e1x.9c1iteGdPsatastheoΓw(4s2)atrpiopwleetrd-leacwretaesmespweriathtuirnecrveaarsiiantgiopnreasnsduries.pTrhoepo1r/tTio1nbaellotowTT5c=at1t.e5m5pKeraattPure=s
p considerablybelowTc,whichindicatestheexistenceofpointnodesinthesuperconductinggap
u function. The data can be well fitted by the gap model ∆(θ)=∆ sinθ with ∆ = 3.08k T .
0 0 B c
s
The relation between the superconductivity and the quadrupole fluctuations associated with
.
t the Γ(2) state is discussed.
a 4
m
KEYWORDS: PrOs Sb , superconductivity, NQR,pressure
4 12
-
d
n
o The filled skutterudite compound PrOs Sb is the residual density of states in the superconducting gap,17
4 12
c first praseodymium (Pr)-based heavy-fermion supercon- but the nodal structure remains unclear.
[ ductor with Tc = 1.85 K.1 The heavy-electron mass has In this paper, we report on the 123Sb-NQR study of
1 been confirmed by the large specific heat jump ∆C/Tc PrOs4Sb12 under pressure. Applying pressure reduces
v 500mJ/(K2mol)at T 1,2 andby de Haas-vanAlphen T 2 andmayalsochange∆ ,18 andthereforecanpro-
c c CEF
3 e∼ffect measurements.3 The ground state of the crystal vide new information on the symmetry of the supercon-
1
electricfield(CEF)foraPr3+ionisaΓ singlet,whichis ductinggapaswellasonthemechanismofthesupercon-
6 1
1 separatedbythefirstexcitedstateoftheΓ(2) tripletbya ductivity. We find that ∆CEF decreases with increasing
4
0 gapof∆CEF 10K.4–9 Becauseofthissmall∆CEF,the pressure, and that at P = 1.91 GPa, 1/T1 decreases in
7 relation betw∼een the quadrupole fluctuations associated proportion to T5 at low temperatures, which indicates
0
with the Γ(2) state5,10 and the occurrence of the super- point nodes in the superconducting gap function.
/ 4
at conductivity has been the focus of discussions.9,11,12 Single crystals of PrOs4Sb12 were grown by the Sb-
m Although the superconducting gap function is im- fluxmethod.ForNQRmeasurements,thesinglecrystals
werepowderedtoallowtherfmagneticfieldtopenetrate
portant for understanding the superconductivity in
-
d PrOs Sb , it has not yet been determined experi- into the sample. However, the size of the grains is kept
4 12
n larger than 100 µm to avoid crystal distortions. Hydro-
mentally.Previousnuclear-quadrupole-resonance(NQR)
o staticpressurewasappliedusingaNiCrAl/BeCupiston-
measurement has revealed the unconventional nature of
c
: the superconductivity.13 The spin-lattice relaxation rate cylinder cell, filled with Si oil as a pressure-transmitting
v medium.19 The pressure at low temperatures was deter-
1/T shows no coherence peak just below T , nonethe-
i 1 c
X less, it follows an exponential temperature dependence minedfromthepressuredependenceoftheTcvalueofSn
metalmeasuredbyaconventionalfour-terminalmethod.
r at low temperatures. On the basis of this result and the
a Data below 1.4 K were collected using a 3He/4He dilu-
exponential decrease in the penetration depth below T
c
foundbymuonspinrelaxation(µSR),14 ithasbeenpro- tion refrigerator at P = 0 and a 3He refrigerator under
high pressure P = 1.91 GPa. To avoid possible heat-
posed that the superconducting gap for PrOs Sb is
4 12
isotropic.13,14Incontrast,oscillationswithrespecttothe ing due to the rf pulses, we used very small amplitude
rf pulses in the T measurements at low temperatures.
magnetic field angle have been found in angle-resolved 1
We confirmed the lack of such a heating effect by ensur-
thermal conductivity measurements, which suggest that
ing that the spin echo intensity is not affected by an rf
thesuperconductinggapisanisotropic,andtheexistence
of point nodes in the gap function has been proposed.15 pulse with a slightly off-resonance frequency, which was
applied before the π/2 π pulse sequence.
The results of penetration depth measurement also sug-
−
gested a point-nodes gap.16 Our previous NQR study The Sb nuclei have two isotopes of 121Sb and 123Sb
with natural abundances of 57.3% and 42.7%, respec-
onthe substitution systemPr(Os1−xRux)4Sb12 strongly
tively. Since 121Sb and 123Sb have the nuclear spins I
suggested the existence of nodes in the superconducting
= 5/2 and 7/2, respectively, five Sb-NQR transitions
gap function of PrOs Sb , since Ru doping at the Os
4 12
are observed.13 In the present experiment, all measure-
site in PrOs Sb as a nonmagnetic impurity induces a
4 12
2 J.Phys.Soc.Jpn. Letter AuthorName
50.00 40 PrOs4Sb12
Hz) 1P23SrOb-sN4SQbR12 (2 Q) T (K)1005 -1-1 cK) 30 , PP == 01 .91 GPa
y (M 49.95 00 Press2ure (GP4a) T (se
nc T1 20
eque , PP == 10 .91 GPa 1 / Tc
Fr 49.90 P = 2.34 GPa 10
T0 0
0 5 10 15 20
49.85
0 5 10 15 20 25 Temperature (K)
Temperature (K) Fig. 2. (color online) Temperature dependence of 1/T1T at P
=0(solidcircles)and1.91GPa(solidtriangles)alongwiththe
Fig. 1. (color online) Temperature dependence of 2νQ position data at ambient P cited from literature13 (open circles). Solid
of123Sb-NQR(±3/2↔±5/2transition)atP =0(solidcircles),
and dotted arrows indicate Tc(P) at P = 0 and 1.91 GPa, re-
1.91(solidtriangles),and2.34GPa(opensquares)alongwiththe
spectively.
dataatP =0(opencircles)byKotegawaetal.13 Solid,dotted,
and dashed arrows indicate T0 at P = 0, 1.91, and 2.34 GPa,
respectively(seetext).Theinsetshowsthepressuredependence
ofT0.
PrOs4 Sb12
2 Tc
ments were carried out at the 3/2 5/2 transition 10
± ↔ ±
(hereafter,2ν transitionforshort)ofthe123Sbnucleus.
Q
Figure 1 shows the increase in the 2νQ resonance fre- 1
quency below T = 25 K for various pressures. T is the 1 )10
0 - c
temperature at which the 2ν resonance frequency in- e
Q s , P = 0
creasesabruptly.Sincetheelectricalfieldgradient(EFG) T (1100 P = 1.91 GPa
itsribpuretdioonm,itnhaentNlyQdRetferremquinenedcybiys athgeooond-smiteeacshuarergoefdtihse- 1 / -1 a.u.) Tc
population of the ground/excited state. Indeed, in both 10 (
PrOs4Sb1213 and PrRu4Sb12,20 T0 is in good agreement c- P=0
with∆CEF/kB.Morerecently,ithasbeensuggestedthat -2 a P=1.91GPa
10
the temperature dependence of NQR frequency can be
0 1 2 3
accounted for by the EFG associated with the hexade-
capole moment of the Γ(2) state.21 Therefore, the in- 0.1 1 10
4
creaseintheNQRfrequencybelowT indicatesthatthe
0 Temperature (K)
depopulation ofthe Γ(2) state occurs below this temper-
4 Fig. 3. (coloronline)Temperaturedependenceof1/T1 atP =0
ature. As seen in Fig. 1, T shifts to lower temperatures
0 (solidcircles)and1.91GPa(solidtriangles)alongwiththedata
T0(P) 7.5 and 5 K at P = 1.91 and 2.34 GPa, re- atambientP citedfromref.13(opencircles).Thestraightlineis
spective∼ly (also see Fig. 1 inset). These results provide aguidetotheeyes.Thedottedcurvedepictstherelation1/T1∝
evidence that ∆ decreases with increasing pressure. exp(-∆0/kBT)with∆0/kBTc =3.45,proposedbyYogietal.23
CEF
Theinsetshowsthetemperaturedependenceofac-susceptibility
The above conclusion is supported by the pressure ef-
atP =0and1.91GPa. Solidanddotted arrowsindicate Tc at
fect on the temperature dependence of 1/T T. Figure 2
1 P =0and1.91GPa,respectively.
shows the temperature dependence of 1/T T at P = 0
1
and 1.91 GPa. The pressure effect appears below 4 K.
At P = 0, the reduction of 1/T1T results in a peak in pendence of ac-susceptibility measured using the NQR
the plot of 1/T1T versus T, which is due to the depopu- coil. Tc decreased from 1.85 K at P = 0 to 1.55 K at P
lation of the Γ(2) state below T . At P =1.91 GPa, the = 1.91 GPa. The main panel of Fig. 3 shows the tem-
4 0
decrease in 1/T T occurs at a lower temperature, indi- perature dependence of 1/T at P = 0 and 1.91 GPa.
1 1
cating the decrease in ∆ . These results are consis- The ambient-pressure data are in excellent agreement
CEF
tentwiththeconclusioninferredfromthemagnetization with those reported previously (open circles),13 except
measurement.18 that 1/T below T 0.4 K is smaller in the present
1
∼
We find concomitantly that the temperature of the sample, probably due to the improvement of the sam-
onset of the superconducting transition decreased with ple quality.22 The datacouldbefitted byanexponential
increasing pressure, in agreement with previous re- function1/T exp(-∆ /k T)belowT ,aspointedout
1 0 B c
∝
ports.2,18 The inset in Fig. 3 shows the temperature de- previously.13 At both P = 0 and 1.91 GPa, 1/T is T-
1
J.Phys.Soc.Jpn. Letter AuthorName 3
independent above T , indicating that the relaxation in is reduced. Clearly, our results do not lend a straight-
0
the high-temperature region is predominated by the Pr- forward support to this theory. Further experimental
4f2-derived localized magnetic moments. With decreas- study under higher pressure is highly desirable. Finally,
ingtemperaturebelowT ,1/T startstodecrease.Below we comment on the different temperature dependences
0 1
T , no coherence peak is observed at T = 1.55 K for P of 1/T at P = 0 and 1.91 GPa. Two possible causes
c c 1
= 1.91GPa,asfor P =0.13 However,the T dependence could be responsible. First, the larger gap ∆ may
CEF
of 1/T at high pressure is markedly different from that contribute to the reduction of 1/T below T at ambient
1 1 c
at ambient pressure.1/T at P = 1.91 GPa decreases in pressure, which makes the temperature dependence of
1
a power law of T below T . 1/T exponential.Second,itmaybeduetothemultiple-
c 1
In particular, below T 0.55 K, 1/T is proportional band nature of the superconductivity.27 Recent thermal
1
∼
to T5, as can be seen more clearly in Fig. 4. We find conductivity measurement under a magnetic field sug-
that a point-nodes model, with a low-energy (E) super- gests the superconductivity at ambient pressure is in-
conducting density of states (DOS) proportional to E2, duced in two different Fermi sheets,27 which may have
can well explain the experimental result. In Fig. 4, the different symmetry. The sheet in which nodes develop
curve below T is a fit to the Anderson-Brinkman-Morel may grow significantly under high pressures.
c
(ABM) model.24,25 Namely, In conclusion, we have presented the 123Sb-NQR re-
sults on the filled skutterudite heavy-fermioncompound
T (T ) 2 N (E) 2 PrOs4Sb12 at P = 0, 1.91, and 2.34 GPa. The tempera-
1 c S
= f(E)[1 f(E)]dE, ture dependence of NQR frequency and the spin-lattice
T k T Z (cid:18) N (cid:19) −
1 B 0 relaxationrate1/T indicatethatthegap∆ between
1 CEF
where NS(E)/N0 = E/√E2−∆2 with ∆(θ) = ∆0sinθ. thegroundstateΓ1singletandthefirstexcitedstateΓ(42)
The fit gives ∆0/kBTc = 3.08. The penetration depth triplet decreases with increasing pressure. At P = 1.91
data at ambient pressure seem to be consistent with our GPa, the temperature dependence of 1/T below T is
1 c
results.16 However, the p+h models26 proposed to ex- well explained by the ABM superconducting state, with
plain the thermal conductivity would give a T3-like de- point nodes in the gap function. To confirm the mecha-
pendence, since the DOS at low-E is linear in E, and is nism showing why T decreases with increasing pressure
c
therefore not compatible with our data. in PrOs Sb , further NQR measurements under pres-
4 12
sure are now in progress.
We thank K. Matano for his experimental help, and
R. Shiina, H. Harima, Y. Kitaoka, H. Tou, M. Ichioka,
N. Nakai, H. Kotegawa,and M. Yogifor discussionsand
comments.Thisworkwassupportedinpartbygrantsfor
2
10 scientific research, and a grant for Attractive Education
Tc in Graduate School ”Training Program for Pioneers of
Frontier and Fundamental Sciences”, from the Ministry
of Education, Culture, Sports, Science and Technology.
1
10
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