Table Of ContentAPS/123-QED
Neutron Scattering Study on the Field-Induced O -type Antiferroquadrupolar
xy
Ordering of Heavy-Fermion Superconductor PrOs Sb
4 12
K. Kaneko,1 N. Metoki,1,2 R. Shiina,3 T. D. Matsuda,1 M. Kohgi,3 K. Kuwahara,3 and N. Bernhoeft4
1Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Naka, Ibaraki 319-1195, Japan∗
2Department of Physics, Tohoku University, Sendai 980-8578, Japan
3Department of Physics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan
7 4DRFMC-CEA, 38054 Grenoble, France†
0 (Dated: February 6, 2008)
0
2 Neutron scattering experiments haverevealed a field-induced antiferroquadrupolar order param-
eterinthePr-basedheavy-fermionsuperconductorPrOs4Sb12. Weobservedthefield-inducedanti-
n
ferromagnetic dipolemomentwith thepropagation vectorq =(100) fortheapplied fielddirection
a
both Hk[110] and [001]. For Hk[110] at 8T, it should be noted that the induced antiferromag-
J
neticmomentof0.16(10) µB/Prorientsparalleltothefieldwithinourexperimentalaccuracy. This
5
observation is strong evidence that the Oxy electric quadrupoleto be the primary order parameter
2
for Hk[110]. A mean-field calculation, based on Γ1 singlet and Γ(42) excited triplet with the Oxy
quadrupolar interaction, reproduces the induced moment direction and its field response. These
]
el facts indicate thepredominance of Oxy-typeantiferroquadrupolar interaction in PrOs4Sb12.
-
r PACSnumbers: 75.40.Cx,75.25.+z,61.12.Ld,74.70.Tx,75.30.Kz
t
s
.
t
a I. INTRODUCTION ferromagnetic dipole order appears simultaneously with
m the coupled order parameters such as quadrupole, oc-
- tupole, and hexadecapole in the order parameter space
d PrOs4Sb12 is the first Pr-based heavy-fermion super- with given T symmetry.18 Since neutrons exhibit no
h
n conductor and attracts considerable interest because of
cross section for an electric quadrupole moment, the ob-
o unusual superconducting properties.1,2,3,4,5,6,7,8,9 One of
servation of the field-induced antiferromagnetic peak by
c the most interesting topics is what is the role of the
[ neutron is not a direct proof for the primary AFQ order
multipole moments of Pr f electrons for the heavy- parameter. Inthis sense,ourpreviousstudy didnotrule
1 fermion nature and unconventional superconductivity in out possibilities of other high-rank multipoles.16 Actu-
v PrOs Sb .10,11 Long range magnetic dipole order does
4 12 ally,itisshownthatthequalitativefeaturesforHk[001],
9 not coexist with superconductivity at zero field.1,2 In-
the induced antiferromagnetic moment perpendicular to
0
stead, the existence of a magnetic field-induced ordered
6 the applied field and the phase diagram can be repro-
phasehasbeenreportedabovetheuppercriticalfieldH
1 c2 duced in terms of octupole as well as quadrupole18,20.
forthesuperconductivity.12,13,14,15 Inourpreviousstudy
0 Recent intensive studies on skutterudites report an im-
7 antiferromagnetic reflections with q=(100) were ob- portant role of higher rank multipole, octupole21 and
0 servedinthe field-inducedorderedphaseforHk[001].16 hexadecapole22, in some compounds. Therefore, it is in-
/ The antiferromagnetic moment of 0.02µ /Pr at 8T
t B dispensable to clarify strictly the dominant interaction
a along the [010] direction perpendicular to the applied
in PrOs Sb . For this purpose, further information is
m 4 12
field was significantly smaller than the field-induced fer-
required: field direction and pressure dependencies and
- romagneticmomentparalleltoH.Thissmalldipolemo-
d inelastic neutron scattering.
ment and the H-T phase diagram can be well explained
n Careful magnetization measurements on PrOs Sb
by the mean-field calculation based on the crystal field 4 12
o clarified that the field-induced ordered phase also exists
c level scheme of the Γ1 singlet ground state with Γ(42) for [110] and [111] with remarkable anisotropy in the
v: tripletexcitedstateandassuminganantiferroquadrupo- H-T phase diagram.14 The existence ofthe field-induced
i lar (AFQ) interaction.16,17,18 The field-induced ordered orderedphaseforthesedifferentfielddirectionsalsosup-
X
phase appearsas aresultof the levelcrossingofthe zero
ports the singlet-triplet crystal field level scheme. It
r fieldgroundandexcitedstatewhichcomesfromthefield
a has been pointed out that the anisotropy in the phase
splittripletstate. Thelevelschemeanditsfieldresponse
diagram is intrinsic to T symmetry and the predomi-
h
have been experimentally clarified by means of neutron nance of quadrupolar interaction.17,18 The field-induced
inelastic scattering under magnetic fields.19 The overall
orderedstates for both Hk[110]and [111]arealso con-
aspect of the field-induced ordered phase indicates that
sidered to be the AFQ ordered phase and not to be the
the antiferromagnetic dipole is not a primary order pa-
dipolar one. The order parameter for Hk[110] was the-
rameter. oretically predicted to be O 18, but has not been con-
xy
Generallyspeaking,therearemanypossibilitiesforthe firmed so far. If this is the case, one could discriminate
primary multipole order parameters. The ordered phase theprimaryorderparameterofnon-magneticquadrupole
is stable only under applied magnetic field, where the from coupled magnetic octupole since the induced anti-
time reversal symmetry has been broken, thus, the anti- ferromagnetic moment direction is expected to be dif-
2
s ) 1.0 PrOs4Sb12 (a)
0 ( 2 -2 l )
0
6 0.29 K H || [1 1 0]
nt / 0.8 0 T
c
3 8 T
0
( 1 0.6 10 T
y
sit
s )en 0.4
0 nt
0I
6
FIG. 1: A photograph of the single crystalline sample of nt / 0.95 1.00 1.05
PrOs4Sb12 used for thepresent neutron scattering study. 3 c l ( r. l. u. )
0
1
(
fiserewnotrtbhewtwheileenitnheomrd.erThteorecfloarreifytheasdtuodmyinfaonrtHinkt[1er1a0c-] nsity 0 s ) 1.0 PHr O|| [s14 S1b 01]2 (b)
e0 ( 2 -2 1 )
mtioangnientizPartOiosn4Smb1e2a.surBeemsiednetss, rietvmeaoletdivatthees euxsisttheantcethoef Intnt / 6 0.27 K
some additional anomalies within the field-induced or- 3 c 0.51 K
dered phase for Hk[110], indicating possible transitions 10 0.70 K
between multipolar ordered states.14 y ( 1.5 K
The purpose of the present neutron scattering study sit 0.5
n
is to clarify the order parameter of the field-induced or- e
deredphase for Hk[110]andto investigatethe originof Int
anomalies in this ordered state. We found that the in- ) 4
s
duced antiferromagnetic moment is parallel to the field 0 208
2 1
dquiraedctriuopno,lewhasicthhiesptrhiemsatrryonogrdeevridpeanracemfeotrert.he Oxy-type 30 cnt / 3 dI / dH 40 (c)
1
II. EXPERIMENT y ( 2 0 2 m40 H6 8 10 PrOs4Sb12
sit ( 1 -1 0 )
en 1 0.27 K
Neutron scattering experiments have been carried out nt
I 0 2 4 6 8 10
on the cold neutron triple-axis spectrometer LTAS in- m
stalled in the guide hall of the researchreactorJRR-3 in 0 H ( T )
Japan Atomic Energy Agency, JAEA. The instrumental
setup was just the same as our previous experiment.16 FIG.2: (a)Thefielddependenceofthe2¯21superlatticere-
The use ofacoldneutronwithtriple-axismode wascru- flectionprofileofPrOs Sb measuredat0.29KforHk[110].
4 12
cialfortakinghighqualitydatatoreducethecontamina- The solid lines denote Gaussian fitting. (b) The field de-
tion of elastic spectra by the low-energy excitations. We pendenceof the 2¯21 reflection intensity measured at various
used a liquid He-free 10T superconducting magnet and temperatures. (c) The 1¯10 reflection intensity plotted as a
3He-4HedilutionrefrigeratorbothdevelopedbyJAEA.23 function of applied magnetic field taken at 0.27K. The inset
Theverticalfieldwasappliedalong[110],perpendicular gives its differential curve, dI/dH.
to the (hh¯l) scattering plane. We observed superlattice
peaks at h+k+l =odd, while ferromagnetic scattering
issuperposedonnuclearBraggpeakath+k+l=even, measurements; we observed clear Curie-Weiss behavior
for example (1¯10). athightemperatureandthemaximumat3.6K,whichis
A large single crystalline sample with a mass of 6g identical to the high-quality sample used in the de Haas
has been grown by the antimony-self-flux method. The van Alphen study.3 The resistivity was also very similar
details of the sample preparation have been published to the previousresults andoursample showedsupercon-
elsewhere.3,24 Figure 1 shows the picture of the sam- ducting transition at the onset of 1.88K. These results
ple, which is composed of small crystallites typically 1 guarantee the high quality of the present sample. For-
mm3 distributed within 1 degree of the crystallographic tunately there was no domain structure concerning the
axes. The four-circle x-ray crystal structure analysis on two-fold symmetry in the basal plane, thus for example,
the small piece of this sample confirms the filled skut- (hkl)reflectionis generallydifferentfrom(khl). We do
teruditestructurewiththespacegroupIm¯3(T5,#204). not know the mechanism for the single domain growth
h
Thesamplewascharacterizedviamagneticsusceptibility process,neverthelessthesingledomainsamplecanbere-
3
15
1.0
)
s (a)
00 PrOs4Sb12 H
nt / 6 0.8 H( 2|| [-12 1 l 0)] 0 0.2.56 KK HAmax
310 c 0.6 8 T 0 1.6.42 KK T ) 10 TA
sity ( H (
en 0.4 m0
nt 5
I
PrOs Sb
0.95 1.00 1.05 4 12
H || [1 1 0]
l ( r. l. u. )
2.0 0
0 0.5 1.0 1.5
) (b)
c
e
S PrOs Sb T ( K )
0 4 12
0 H || [1 1 0]
s / 12 1.5 ( 2 -2 1 ) FTIhGe.c4lo:seTdheanHd-Toppehnacseircdlieasgraarme fioeflPdsrOcso4rSrebs1p2ofnodrinHgkt[1o1t0h]e.
nt 3 T
c onset and the maximum of the field-induced antiferromag-
30 8 T netic peak, respectively. The closed triangles are the phase
( 1 1.0 10 T boundary determined from the temperaturedependence.
y
sit
n
e
nt measuredatvarious temperatures as shownin Fig. 2(b).
I
0.2 0.4 0.6 0.8 1.0 Applying the field at 0.27K, the peak intensity appears
at5T (=H ),reachesa maximumaround8T (=H ),
A max
T ( K )
anddecreasessteeplyabove8T.H increasesasthetem-
A
perature rises: 6T and 7T at 0.51K and 0.71K, respec-
FIG.3: (a)Thetemperaturedependenceofthe2¯21superlat- tively. In contrast, the maximum around 8T is almost
ticereflectionofPrOs4Sb12measuredunderthemagneticfield temperature independent. No trace of superlattice peak
of 8T applied along the [110] direction. The solid lines de- was found above 1K.
notetheresultoffittingwithGaussianandlinearbackground
Figure2(c)displaysafieldvariationofthepeakinten-
foreachtemperature. (b)Thetemperaturedependenceofthe
2¯21reflection intensitymeasured at variousfields. Thesolid sityat0.27KtakenatQ=(1¯10). Anincreaseof1¯10re-
lines are guides for theeyes. flection intensity originating from the uniform magnetic
moment was clearly observed. The inset gives the differ-
ential curve, dI/dH, which corresponds to the differen-
producibly obtained with our sample growth condition, tialcurveofthesquareoftheferromagneticmomentM2.
which is rather mysterious. With applied fields, the intensity exhibits a gradual in-
creaseandchangesits slopearound5.5T and8T,which
can be clearly seen in the inset. These inflection fields
III. RESULTS are consistent with the onset (H ) and maximum fields
A
(H ) in the superlattice reflection intensity as well as
max
Figure2(a)showsthe representativedataforthe scat- the result of magnetization.14. The small hump around
tering profile of the 2¯21 superlattice peak measured at 1.5Tmightarisefromthe breakingofthe heavy-fermion
0.29 K with magnetic fields along the [110] direction. superconductivity since it is veryclose to the upper crit-
An application of the magnetic field of 8T induces a ical field Hc2.
clear resolution-limited superlattice reflection at (2¯21) The temperature dependence of the 2¯21 superlattice
where no trace of peak was observed at zero field. The peak profile under the applied field of 8T is shown in
observedfield-inducedsuperlatticepeakpositionsarethe Fig. 3(a). The 2¯21 superlattice reflection becomes weak
sameasthoseforHk[001],namelythepropagationvec- withoutsignificantpeakshiftand/orbroadeningastem-
tor is q=(100). A marked decrease to ∼1/10 of the perature increases, and no trace of peak was observed
integrated intensity occurs between 8 and 10 T whereas at 1.4K. Figure 3(b) shows peak intensity at (2¯21) as
the magnetic Bragg peaks maintain their positions and a function of temperature taken under the field of 3, 8
widths. and 10T. With increasing temperature from 0.27K at
The field dependence of the 2¯21 peak intensity was 8T, the peak intensity showed the monotonous decrease
4
PrOs Sb
4 12 8 T
0.20 0.29 K 10 T
)
B
m 0.15
(
f |
n 0.10
si
m
| 0.05
H || [1 1 0]
0
1 3 3 1 1
0 0 1 1 2
0 0 1 - 1 - 2 -
H
FIG. 5: The product of the magnitude of the magnetic mo-
mentandtheanglefactorderivedfromtheobservedmagnetic
Bragg reflection intensity. The open and closed circles indi-
cate the data for µ H =8T and 10T, respectively. The dot-
0
ted lines are the fitting results for the reflection in the (hh¯l)
FIG.6: Theobtainedmagneticstructureforthefield-induced
scattering plane by assuming the antiferromagnetic moment
orderedphaseofPrOs Sb withapplication ofthemagnetic
4 12
parallel to theapplied field, namely, sinφ=1. field along Hk[110].
and disappeared around T =0.8K. The feature for 10T
A between the ordered magnetic moment and the scatter-
is quite similar to that for 8T except for the absolute
ing vector Q. The square of the product of angle factor
intensity. No peak was observed in the result for 3T.
andantiferromagneticmomentµcanbegivenasfollows;
Noadditionalanomalyat8Tand10Tisconsistentwith
the fact that H is almost temperature independent.
max
I
The present results for Hk[110] of PrOs4Sb12 are sum- |µsinφ|= mag .
marized in the H-T magnetic phase diagram as shown sKL(θ)[0.269×10−12f(Q)Fmag(Q)]2
in Fig. 4. The closed and open circles are the field cor- (2)
responding to the onset and the maximum of the field- Figure 5 shows the square root of integrated intensity
inducedantiferromagneticpeak,respectively. Theclosed divided by scale factor, Lorentz factor, structure factor
triangles are the phase boundary determined from the andmagneticformfactorofthe Pr3+ freeion. Therefore
temperature dependence of the antiferromagnetic inten- the verticalaxis in Fig. 5 is the product of the antiferro-
sity. The overall aspect of the phase boundary, deter- magneticmomentµandtheanglefactorsinφ. Asclearly
mined from the present diffraction experiments, is con- seeninFig.5,thisquantityisalmostisotropicwithinthe
sistentwiththeresultofmagnetizationmeasurements.14 present scattering plane. Although the data obtained at
The field-induced superlattice reflection with q=(100) 10T is much weaker, it is also isotropic. This isotropic
exists only in the region of the field-induced ordered nature of the left hand side of equation Eq.(2) indicates
phase,inotherwords,theorderedstatewithq=(100)is that the antiferromagnetic moment is perpendicular to
the basisof the field-induced orderedphase for Hk[110] the (hh¯l) scattering plane at both 8T and 10T within
which is the same as Hk[001]. ourexperimentalaccuracy. Thus the induceddipole mo-
In order to determine the magnetic structure in the ment observed by neutron is parallel to the direction of
field-induced ordered phase, the integrated intensity of the applied field along [110]. Using this angle factor,
thesuperlatticepeaksonthe(hh¯l)scatteringplanewere sinφ=1, the induced antiferromagnetic moment for 8T
measured. The magnetic Bragg reflection intensity for and10TisdeducedtobeµAF =0.16(10)µBand0.07µB,
unpolarized neutron diffraction can be written, respectively. The antiferromagnetic moment exhibits a
strong reduction in the magnitude from 8T to 10T.
I (Q)=KL(θ)[0.269×10−12f(Q)(µsinφ)|F (Q)|]2 Thereisalargeinducedferromagneticmomentparallel
mag mag
(1) tothefield,whichisroughlytentimeslargerthanthatof
where K is the scale factor,L(θ) is the Lorentz factor,µ the field-induced antiferromagnetic moment.14 The sum
is the antiferromagnetic moment, f(Q) is the magnetic oftheferromagneticandantiferromagneticmomentgives
form factor and F (Q) is the magnetic structure fac- themagneticstructureinthefield-inducedorderedphase
mag
tor which is unity for the present case. φ is the angle as shown in Fig. 6. The moment size is different for
5
theeachmagneticsublatticeduetotheantiferromagnetic
2.0
component parallel to the field direction; the magnetic
moment at the corner of the unit cell is larger than that 1.5 PrOs4Sb12
H || [1 1 0] 1.5
at the center, or vice versa.
)B 1 -1 0
m
( 1.0 1.0
IV. MEAN-FIELD ANALYSIS M F M
0.5 0.5
c
a
sulWtsewpirthesHentk[a1m10e]a.nI-nfiePldrOcsalScublat,iothnefoΓr tshinegplertesgernoturned- 0 lc (
statewiththeΓ(42) firstexcite4dtr1i2pletsta1teat8Kiswell PrOs4Sb12 0.2 B m
tshepaatrtahteedlofwro-lmyinthgesointghleert-etxrcipitleedtlsetvaetles.1a6r,2e5r,2e6spTohnissimbleeafnosr m ) B 0.10 H 1|| [-11 11 0] )
lowtemperature physicalproperties. Hence,we moveon (
our discussions to this low-lying singlet-triplet subspace. F 0.05 0.1
A
Activemultipolesforthesinglet-tripletsystemwithJ=4 M
in the O and T symmetries are classified in ref.17 and
h h
18. According to the symmetry lowering from O to T ,
h h 0 0
the triplet states in T are represented by linear com-
h 0 5 10
m
binations of those in O , whereas the Γ singlet is not
h 1 0 H ( T )
influenced. The mixing is parameterizedby y,the coeffi-
cientinthe crystalfieldHamiltonianreflectingthe effect
of T , where y =0 corresponds to O .27 FIG. 7: The field dependenceof the field-induced ferromag-
h h
neticandantiferromagneticmoment. Theopencirclesareob-
The possible multipole order parameters under mag-
tained from theneutrondiffraction, whose valuecorresponds
netic fields which cause admixture are classified by sym-
totheleftaxes. Theresultsofmean-fieldcalculationarerep-
metry analysis. There are two possible symmetries for
resented by lines corresponding to theright axes.
Hk[110],Γ andΓ . ForΓ ,themagneticdipoleJ +J
1 2 1 x y
is mixed with J −J depending on the magnitude of
x y
y. The induced antiferromagnetic moment in the field- where ∆ is the singlet-triplet energy splitting and h is
induced ordered phase was revealedto be J +J in the
x y a scaled magnetic field. The appearance of an effective
present experiment. This result indicates the order pa-
staggered field (δ) to the two spins is characteristic of
rameterforHk[110]tobetheΓ symmetrywhichiscon-
1 the Th system. It has been shown that the interaction
sistent with the theoretical prediction.18 Furthermore,
of O -type quadrupoles is mapped to the biquadratic
xy
the induced moment parallel to applied fields within the
pseudo-spin Hamiltonian,
experimental accuracy suggests the small y parameter
whichisalsoconsistentwiththe previousresultsonneu-
H =D 4 (s ×s )·(s ×s )+ǫ µ ·µ
tron diffraction for Hk[001] and a small anisotropy in Q Q 1i 2i 1j 2j 1 i j
the H-T phase diagram. X(ij) h
Hereafter, we study to what extent the simple +ǫ2 (s 1i×s 2i)·µ j +µ i·(s 1j ×s 2j) , (5)
quadrupolarinteractionmodelcanreproducethepresent
(cid:16) (cid:17)i
observed results. Our analysis is based on the Hamilto- where
nian consisting of crystal field potential, Zeeman energy
and AFQ interaction, µ =(sy1sz2+sz1sy2, sz1sx2 +sx1sz2, sx1sy2 +sy1sx2). (6)
See ref. 17 and 18 for details on the dependence of ǫ
H =H +H +H . (3) 1
CF Z Q
and ǫ on the T parameter y. An advantage of this
2 h
mapping is to clarify thatthe predominantandisotropic
It is shown that the mechanism of field-induced AFQ
contribution to the field-induced order is given by the
order in this model can be clarified with a pseudo-spin
representation analogous to the dimer-spin systems.17,18 partofavectorproducts 1×s 2,whereastheremaining
part related with µ produces an anisotropic correction
With the use of two pseudo-spins, s and s with S =
1 2
due to the T symmetry.
1/2,whicharedefinedinthesinglet-tripletsubspace,the h
In the following, we discuss the mean-field solutions
local part of the Hamiltonian is described as
of this model in the field direction Hk[110], in which
the stable phase is shown to be of O . The calculation
3 xy
HCF + HZ =∆ +s 1i·s 2i has been carriedout by using the same parameter set as
4
i (cid:18) (cid:19) in ref. 18. Namely, we assume D z = 0.3∆, ∆ = 8K,
X Q
− h · s +s +δ(s −s ) , (4) x=0.45andy =0.12,wherezisthenumberofthenear-
1i 2i 1i 2i
estneighborbonds,and(x,y)arethecrystalfieldparam-
i
X(cid:0) (cid:1)
6
π / 2 interaction. Thus,itis quite instructive to seethe differ-
( H ) ence between quadrupolar and magnetic interactions by
[1 1 0] ( || H )
z interpolating the two limits.
m
n Intheframeworkofthemean-fieldtheory,wehavecal-
o z
cti AF culated the antiferromagnetic moment direction for the
dire π / 4 [0 0 1] totalHamiltonianH+HMinthefielddirectionHk[110].
nt Taking the ratio of the interactions as DQ = D(1−η)
me andDM =Dη,theanglefromtheappliedfielddirection,
o [1 1 0] ζ, is plotted in Fig. 8 as a function of relative magnetic
m
strength η = D /(D +D ) with a fixed Dz(= 0.3∆).
AF PrOs4Sb12 In the case thaMt theQmagnMetic interaction is small, the
H || [1 1 0]
0 momentdirectionisalmostparalleltothefield. Whereas
( || H )
themagneticinteractionbecomesdominant,themoment
0 0.5 1.0
h orients perpendicular to the field, as expected in the
( D / ( D + D ) ) dimer-spin systems. Such a remarkable change of the
M Q M
antiferromagnetic structure that depends on the inter-
FIG.8: Calculatedangleζ betweenthefield-inducedantifer- action is characteristic in the field direction Hk[110].
romagneticmomentandthefielddirection[110]asafunction In the case of Hk[001], both interactions result in the
of relative magnetic interaction strength η for h/∆=1.2, cor- antiferromagnetic moment perpendicular to the field.30
responding to µ0H≃7T. Thus, the present experiment for Hk[110] is crucial in
distinguishingthetypeofmultipolarinteractions. Inthis
sense,theobservedantiferromagneticmomentparallelto
eters. Theresultsofmagneticfielddependenceofthefer-
the field is regarded as strong evidence of the predomi-
romagnetic (M ) and antiferromagnetic moment (M )
F AF nant AFQ interaction.
areshowninFig.7. Theyarecomparedwiththepresent
experimentalresultsthatarededucedbynormalizingM
F
and M at 8T to be 1.2µ 14 and 0.16µ , respectively.
AF B B V. DISCUSSION
Onefindsthattheobservedoverallfieldresponseofboth
ferromagneticandantiferromagneticmomentsiswellex-
The field-induced antiferromagnetic moment parallel
plained by this model analysis. These agreement clearly
to the field for Hk[110] is different from the perpen-
indicates that the staggered moment should be induced
dicularantiferromagneticmomentforHk[001],whereas
by the primaryAFQ orderwhichremainsinaccessible to
the propagationvector q=(100) is the same. In the lat-
the neutron probe. It is also shown that the calculated
ter case the antiferromagnetic moment is parallel to the
value of antiferromagnetic moment is larger than that
[010] direction, and not to the [100] direction; a clear
for Hk[001], and this tendency is qualitatively consis-
peakwasobservedfor q=(100)butno traceofthe anti-
tentwiththeobservedresults. Note,however,thatthere
ferromagnticpeakwasobservedforq=(010). Thischar-
existquantitativedifferencesbetweentheoryandexperi-
acteristicmomentdirectionreflectingthelackoffour-fold
ment, probablydue to fluctuationeffects orcontribution
symmetry gives the important information on the possi-
of other multipole interactions.
ble order parameter. Variation in the orientation of the
In order to clarify the unique properties of the AFQ
field-inducedantiferromagneticmomentaccordingtothe
model more definitely, let us study the effects of pos-
appliedfielddirectionischaracteristictothequadrupolar
sible magnetic interaction. It is shown that the mag-
ordered phase. Both results indicate the predominance
netic interactionof dipoles and octupoles is expressedas
an isotropic bilinear spin Hamiltonian.18 Although the of the Oxy-type AFQ interaction in PrOs4Sb12.
genericformoftheinteractionissomewhatcomplexwith A recent inelastic neutron scattering study clarified
afewparameters,weintroduceherethesimplestformto thatthecrystalfieldexcitationsfromtheΓ1groundstate
capture the essential physics, to the Γ(2) triplet first excited state have a dispersion;
4
the observed excitation softens at q = (100) which is
H =D (s ·s +s ·s ). (7) thesameastheorderingvectorofthefield-inducedAFQ
M M 1i 1j 2i 2j
ordered state.25 In addition, the intensity at q=(100)
X(ij)
is weaker than that at zone center. Though the mag-
This symmetric model in the two spins corresponds to a netic and non-magnetic interaction gives the same en-
mixed interaction of dipoles and octupoles. Note, how- ergy dispersion, the q dependences of the intensity are
ever, that the relevant part is the octupolar interaction, opposite;29 namely, magnetic interaction should lead to
because they have much larger matrix elements between the stronger intensity at the zone boundary. Thus, the
singlet and triplet states. It should be stressed that the observedexcitationspectraindicatethedominanceofan-
total Hamiltonian in the limit, D =0 and D 6=0, re- tiferroquadrupolar interaction which is quite consistent
Q M
duces to the conventional dimer-spin model28, in which with the present result for Hk[110]. Together with the
a similar field-induced ordertakes place by the magnetic anisotropyintheH-T phasediagram,the predominance
7
of AFQ interaction is clarified in PrOs Sb . moment direction from [110]kH to [001]⊥H should be
4 12
Furthermore, the excitation observed in the inelastic accompanied by the quadrupole-quadrupole transition.
scatteringstudy givesaninteractionstrength. Notethat This reorientation of the antiferomagnetic moment has
the quadrupolarcoupling constantobtainedfromthe in- not been observed in our present neutron scattering ex-
elastic scattering is almost consistent with that used in periment up to 10T. In order to clarify the origin of the
the present study. There is a slight difference in the def- anomalies observed in magnetization, higher field neu-
inition; D in the present paper can be connected with tron scattering experiments should be carried out.
Q
dQ in ref.25with DQ =β2dQ where β is the off-diagonal In conclusion, we observed the field-induced anti-
matrix elements of quadrupolar moments between sin- ferromagnetic order of heavy-fermion superconductor
glet and triplet. Taking this difference into account, the PrOs Sb forHk[110]. Theantiferromagneticmoment
4 12
coupling constant β2dQz≃-2.7K derived from inelastic was parallel to the field direction, which is direct evi-
spectraisconsistentwithDQz≃2.4Kusedinthepresent dencefortheprimaryquadrupolarorderparameterofthe
study. In addition, the quadrupolar coupling constant field-induced ordered phase. Together with our previous
′
gΓ4 obtained from the ultrasound measurements of C44 study for Hk[001], we can conclude that the Oxy-type
isalsoconsistentwiththepresentvalue.8 Inotherwords, quadrupolar interaction is intrinsic to the many body
the coupling constant used in the present study corre-
interaction between f electron based quasiparticles in
sponds with those obtained in the other measurements. heavy-fermion superconductor PrOs Sb .
4 12
The maximum in the induced antiferromagnetic mo-
ment, H , corresponds well to the H anomaly in the
max 1
magnetization14,asshowninFig.4. However,nochange
in the antiferromagnetic structure was found on pass-
Acknowledgments
ing through H . As shown in Fig. 7, the induced
max
J +J appears to vanish slightly above 10T. This crit-
x y
ical field is well below the reported critical field for the We areverygratefultoJ.-M.MignotandA.Gukasov
field-induced ordered phase of ∼12T but rather close to forcommunicationofunpublished dataobtainedatLLB
the H anomaly.14 Inother words,there is anotherfield- Saclay. We also wish to thank H. Sugawara for helpful
2
induced ordered state above H . A possible explanation advice on sample preparation. This work was supported
2
wouldbetheswitchingoftheorderparameterfromΓ to byGrants-in-AidforScientificResearch,YoungScientist
1
Γ representationatH . Itistheoreticallypredictedthat (B) (No. 16740212)and in Priority Area ”Skutterudite”
2 2
the energy difference of these two ordered states is very (No. 18027012,18027013and 18027015)of the Ministry
small.18 Thecoupledmagneticdipoleorderparameterof of Education, Culture , Sports, Science and Technology,
Γ representation is J for Hk[110]. Thus a change of Japan.
2 z
∗ Electronic address: [email protected] 8 T.Goto, Y.Nemoto,K.Sakai,T.Yamaguchi,M. Akatsu,
† Presentaddress: 18MaynestoneRoad,SK236AQ,United T. Yanagisawa, H. Hazama and K. Onuki, H. Sugawara,
Kingdom H. Sato, Phys.Rev.B 69, 180511(R) (2004) .
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8
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20 R. Shiina, unpublished. 30 In the case of the O symmetry (y = 0), the dipoles
h
21 M. Yoshizawa, Y. Nakanishi, M. Oikawa, C. Sekine and I and the octupoles in the singlet-triplet system are sim-
Shirotani, S. R. Saha, H. Sugawara and H. Sato, J. Phys. plyexpressed with thepseudospinsas J ∝s +s and
1 2
Soc. Jpn. 74, 2141 (2005). T β ∝s −s .18WhentheT termisintroduced(y6=0),
1 2 h
22 T. Takimoto, J. Phys.Soc. Jpn. 75, 034714 (2006). they are weakly mixed with each other. Concerning the
23 S. Katano, N. Minakawa, N. Metoki, T. Osakabe, J. mean-fieldsolution of theinteraction (7),it is knownthat
Suzuki, Y. Koike and Y. Ishii, Appl. Phys. A 74, S270 theantiferromangetic orderparameteristransversetothe
(2002). fielddirection28.Iftheycomponentofthespins,orequiva-
24 K. Kaneko, N. Metoki, T. D. Matsuda and M. Kohgi, J. lentlythatofthedipolesandtheoctupoles,ischosenasthe
Phys. Soc. Jpn. 75, 034701 (2006). orderparameterinH k[001],onecanexpecthsyi∼−hsyi
1 2
25 K.Kuwahara,K.Iwasa,M.Kohgi,K.Kaneko,N.Metoki, dueto the positive ∆ term in eq. (4). This leads to a tiny
S. Raymond, M.-A. M´easson, J. Flouquet, H. Sugawara, antiferrmagnetism because Jy ∼ sy1 +sy2 for a small Th
Y. Aokiand H.Sato, Phys. Rev.Lett. 95, 107003 (2005). term, which is compatible to the experimental results for
26 N.A.Frederick,T.A.Sayles,andM.B.Maple,Phys.Rev. H k[001]. Since the model is isotropic for the field direc-
B 71, 064508 (2005). tions,thistransverseantiferromagnetismdoesnotaccount
27 K. Takegahara, H. Harima and A. Yanase, J. Phys. Soc. for thepresent experimental results in H k[110].
Jpn. 70, 1190 (2001); ibid. 70, 3468 (2001); ibid. 71, 372
(2002).
