Table Of ContentMagnetic phase diagram of antiferro-quadrupole ordering in HoB C
2 2
∗
Tatsuya Yanagisawa and Terutaka Goto, Yuichi Nemoto
Graduate School of Science and Technology, Niigata University, Niigata 950-2181, Japan
Ryuta Watanuki
Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan.
5
0 Kazuya Suzuki
0 Graduate School of Engineering, Yokohama National University, Yokohama 240-8501, Japan
2
n Osamu Suzuki and Giyuu Kido
a National Institute for Materials Science, Sakura, Tsukuba 305-0003, Japan
J (Dated: February 2, 2008)
0
2 The magneticphasediagram for antiferro-quadrupole (AFQ)ordering in tetragonal HoB2C2 has
been investigated by measurements of elastic constants C11, C44 and C66 in fields along the basal
x-y plane as well as the principal [001]-axis. The hybrid magnet (GAMA) in Tsukuba Magnetic
]
l Laboratory was employed for high field measurements up to 30 T. The AFQ phase is no longer
e
observed above 26.3 T along theprincipal [001] axis in contrast to therelatively small critical field
-
r of 3.9 T in fields applied along the basal [110] axis. The quadrupolar intersite interaction of Oxy
st and/or O22 is consistent with the anisotropy in the magnetic phase diagram of the AFQ phase in
t. HoB2C2.
a
m PACSnumbers: 71.25.Ld,71.55.Ak,62.65.+k
-
d
n I. INTRODUCTION transition at the Dy3+ LIII absorption edge.14,15 The (0
o 0 l/2) reflections at E2 channels of scatteting suggests
c the anisotropic charge distribution below T is consis-
[ 4f-electron systems, with orbital as well as spin de- Q
tent with the AFQ ordering in basal plane. It was also
grees of freedom, in rare earth compounds frequently
1 reportedthatananisotropicchargedistributiondueto a
show electric quadrupole ordering in addition to mag-
v small buckling lattice distortion of the B- and C- atoms
neticdipoleorderingatlowtemperatures. Wenotedthat
5 contributes to the main peak of the resonantx-rayspec-
0 inHoB6,withaΓ5 tripletgroundstate,ferro-quadrupole traofE1transitioninDyB C .16,17,18,19,20 Theseresults
5 ordering is accompanied by a structural change from 2 2
suggestthat the O - and/orO2-type quadrupole order-
1 a cubic lattice to a trigonal one.1,2,3 CeB , with a Γ xy 2
6 8 ing possessing a charge distribution in the basal plane is
0 quartet ground state, is well known as a typical ex-
5 ample of antiferro-quadrupole (AFQ) ordering in a cu- the order parameter of phase II of DyB2C2.
0
biccompound.4,5,6Thetetragonallanthanidecompounds Onodera et al. reported that an isomorphous com-
/
t DyB C , HoB C and TbB C are also known to show pound HoB C shows an incommensurate short range
a 2 2 2 2 2 2 2 2
m AFQ ordering in competition with antiferro-magnetic magnetic ordered phase IV at TC1 = 5.9 K and succes-
(AFM) ordering.7,8,9 These systems have the tetragonal sivelyundergoesanAFMorderingatT =5.0Kinzero
- C2
d LaB2C2-typestructure10,11,12 withspacegroupP4/mbm field.8 The magnetic structure below TC2 on HoB2C2 is
n as shown in Fig. 1(a). The specific heat and magnetic essentially the same as that of DyB C (Fig. 1(b)). The
2 2
o susceptibility measurements performed by Yamauchi et intermediatephaseIVpossessesalongperiodicmagnetic
c
al. revealed that DyB C exhibits quadrupole order- structure characterized by a propagation vector of k =
v: ing of the Dy3+ (J =2125/2) ions in phase II below (1+δ,δ,δ′)withδ =0.11andδ′ =0.04,alongwithbroad
i T = 24.7 K, which successively transits to an AFM diffuse magnetic scattering around k = (100).21,22 The
X Q
state in phase III at T = 15.3 K.7 Neutron diffrac- phase IVofHoB C is contrastto the absenceofthat in
N 2 2
r
a tion measurements have shown the characteristic AFM DyB2C2. It is noted that neutron scattering on TbB2C2
structure,withaslightlytiltedangle,lyinginthetetrag- and ErB C also show long periodic magnetic order-
2 2
onalbasalplane.11,12 Thetiltingofmagneticmomentsin ing with nearly the same periodicity as HoB C .23,24
2 2
the phase III are attributed to the competitive inter-site We show, in Table I, a dyad of the phase classifica-
interactions between the AFM and AFQ moments. Fur- tion and these properties on HoB C . Here, phase I is
2 2
thermore,fieldinducedAFMmomentsoftheAFQphase para-magnetic (para-quadrupole) phase. The phase III’
II inthe basalplane weredetected byneutronscattering and III” are sub-phase of phase III, which will be dis-
under magnetic fields applied along the [100]-axis.13 Re- cussed later. Some reference with superscript indicated
cently, Tanaka et al. performed detailed resonancex-ray thatthephaseidentifyingbyanalogyfromtheresultson
scattering measurementat two resonantenergies of elec- DyB2C2 and Ho1−xDyxB2C2 . Elastic constants repre-
tric dipole (E1) transition and electric quadrupole (E2) senting quadrupole susceptibility of the 4f-electron sys-
2
TABLE I: Classification and summary of the phase on HoB2C2.
Phase Properties Reference
I Para-magnetic, Para-quadrupole 8,29
a b
II Antiferro-quadrupoleordering (with field induced magnetic moments) (13,14,15,16,17,18,19) ,20, 27
a b
III Antiferro-quadrupole+Antiferro-magnetic ordering (coexistence) 20,21,26 ,27
a
III’ Antiferro-quadrupole+Antiferro-magnetic ordering (with magnetic domains) 26
a a
III” Antiferro-quadrupole+Antiferro-magnetic ordering (with partly reoriented magnetic moments) 7 ,25
IV Incommensurate magnetic ordering (with diffuse neutron scattering) 21,22
aResultsonDyB2C2.
bResultsonHo1−xDyxB2C2.
tem is a useful probe for examining a ground state with
orbital degeneracy or pseudo-degeneracyin particular.28
WehaveperformedultrasonicmeasurementsonHoB C
2 2
and in Fig. 2 we present the measured elastic constants
for comprehension.29 Considerable softening of 22 % for
C below100K,5.5%forC below50K,and2.4%for
44 66
(C −C )/2 below 30 K down to T =5.0 K indicate
11 12 C1
apseudotripletgroundstateconsistingofE-doubletand
A(orB)singletinHoB C .29 InphaseIV,alltransverse
2 2
modes show enhanced softening associated with consid-
erableultrasonicattenuation,whereslowrelaxationtime
of τ ∼7×10−9 s was found.
Themagneticfield-temperature(H-T)phasediagrams
of DyB C and HoB C show anisotropic behavior de-
2 2 2 2
pending on the field directions.8,25 The order parame-
ter of the AFQ phase and its relation to the anisotropy
of the H-T phase diagram still remain to be solved.30
The H-T phase diagram of the AFQ phase II of tetrag- FIG.1: The crystalstructureofHoB2C2. (b)Themagnetic
structureapplyingH k[100]inphaseIIIofHoB2C2,according
onal DyB2C2 and HoB2C2 compounds are often com- to Ohoyamaet al.21 and Zaharkoet al.26
pared to that found in cubic CeB 4,5,6 and La-diluted
6
systems CexLa1−xB6.31,32,33 It has been reported that
CexLa1−xB6 (x=0.75∼0.60)exhibitsanorderedphase
IV being closely to the AFQ and AFM phases. A huge
softening in a transverse elastic constant C and a trig-
44
onallatticedistortioninthephaseIVofCexLa1−xB6 in-
dicatesaspontaneousferro-quadrupolemoment.34 Kubo
and Kuramoto have recently proposeda plausible model
based on octupole ordering to explain the trigonal dis-
tortion in phase IV.35 The order parameter of phase
IV in the present compound HoB C is not settled yet.
2 2
ThecompetitionbetweenAFQandAFMorderinginthe
tetragonal HoB C system is an important issue in the
2 2
present work.
We haveperformedultrasonicmeasurements onsingle
crystals HoB C under magnetic fields in order to inves-
2 2
tigate the anisotropic behavior of the AFQ phase II in
the H-T phase diagram in fields along the three princi-
pal axes H k [110], H k [100] and H k [001]. In Sec.
II the experimental procedure is described. The experi-
FIG.2: Elastic constantsof C44,C66 and (C11−C12)/2 as a
mental results of the elastic constants and the magnetic
function of temperature below 80 K.
phasediagramsarepresentedinSec. III.Theconcluding
remarks are in Sec. IV.
3
FIG.3: Relativechangeoftheelasticconstant∆C66/C66 vs. FIG. 4: Relative change of the elastic constant ∆C66/C66
temperature under various magnetic fields applied along the vs. magnetic field at frequencies of 31 MHz under various
[110]-axis of HoB2C2. Transverse modes at frequencies of 31 temperatures in HoB2C2. Magnetic fields up to 8 T were
MHz were used for themeasurements. applied along the [110]-axis.
II. EXPERIMENTAL DETAILS by the transverseultrasonic mode propagatingalongthe
[100]-axis with polarization along the [010]-axis, corre-
HoB C single crystals were grown with a tetra-arc sponding to a symmetry strain ε with a B(Γ ) repre-
2 2 xy 2
furnace. Rectangularsampleswithdimensionof3×3×2 sentation. The transverse C exhibits a softening of 3
66
mm3and2×2×3mm3werepreparedbyawiredischarge % with decreasing temperature down to the transition
cutterfortheultrasonicmeasurement. Theorientationof temperature T = 5.0 K in zero field. The softening of
C2
crystalwithrespecttoappliedmagneticfieldwassettled C in the phase IV is suppressed in a magnetic field of
66
with in the accuracy of 1 degree by using x-ray Laue 0.5T,andT oftheI-IVtransitionpointshiftstolower
C1
diffractions. The LiNbO transducers for the generation temperatures with increasing fields. The C increases
3 66
anddetectionofthesoundwaveswiththefrequencies8.5 below T being the transition point from the phase IV
C2
MHz and its overtone 31 MHz were bonded on opposite to the phase III. Then these I-IV and III-IV transition
surfacesofthesample. Anultrasonicapparatusbasedon points cross each other at a tetra critical point in H-
the phase comparison method, detecting time-delay for T phase diagram (See Fig. 6 (c)). The minima of C
66
successive ultrasonic echo signals, was used to measure in fields above 1.5 T and up to 3 T indicates transition
the sound-velocity v. For the estimation of the elastic from paramagnetic phase I to the AFQ phase II. Above
constant C = ρv2, we used the mass density ρ = 6.965 4 T, no indication of this phase transition was observed.
g/cm3 from the lattice parameter a=b=0.534 nm and
c=0.352nmofHoB2C2. A3He-evaporationrefrigerator Figure 4 represents the relative change of the elastic
was used for the low-temperature measurements down constantC asafunctionofmagneticfieldforH k[110]
66
to 0.5 K. Magnetic fields up to 12 T were applied by at several temperatures. In temperatures below 4.5 K
a superconducting magnet. Magnetic fields above 12 T anddownto 0.55K,the sharpminima around3.5∼4T
were generated by a hybrid magnet (GAMA) consisting suggest the occurrence of an I-II phase transition, which
ofthesuperconductingmagnetandwater-cooledresistive isexpectedintheorderedphasewithsymmetrybreaking
magnet in Tsukuba Magnet Laboratory (TML) of the character. The intermediate region of two transitions,
National Institute for Materials Science (NIMS). from 2 T to 4 T in H k [110], indicates the AFQ phase
II. The small anomalies at fields lower than 1.9 T are
an indication of the II-III phase transition. The III” -
110
III. RESULTS AND DISCUSSION III phase transitions has been also observed at 1.7 T. A
broad minimum at 6.5 K indicates no sign of the field
A. Magnetic field dependence induced phase transition.
for H k [100] and H k [110] The relative change of the elastic constant C as a
44
functionofmagneticfieldforH k[100]isshowninFig.5.
Figure 3 represents the relative change of elastic con- The C was measured by a transverse ultrasonic mode
44
stant C as a function of temperature under various propagating along the [100]-axis (or [010]-axis) with po-
66
magnetic fields for H k [110]. The C was measured larization along the [001]-axis. The C mode induces a
66 44
4
B. Magnetic field dependence for H k [001]
Figures7(a)and(b)representthetemperaturedepen-
dence of ∆C /C under various fields up to 8 T along
66 66
the principal [001]-axis and field dependence at various
temperatures, respectively. In Fig. 7 (a), the I-IV tran-
sitions at T = 5.9 K, indicated by down arrows, shift
C1
to lower temperatures with increasing field up to 3 T
while the IV-III transition T , indicated by upward ar-
C2
rows, shift slightly to lower temperatures in field. Even-
tually T and T cross each other at 3 T. At 3.5 T,
C1 C2
two anomalies indicating the successive transitions I-II
and II-III were observed. The I-II transition has been
found from 4 T to 8 T in Fig. 7 (a). These transition
points are shown in the phase diagram in Fig. 11. We
successfullyobservedthe AFQ phaseII inhighmagnetic
field applied along H k [001] above 4 T.
Magnetic field dependence of ∆C /C at tempera-
FIG. 5: Relative change of the elastic constant ∆C44/C44 66 66
tures from 0.55 K to 6.5 K is shown in Fig. 7(b). Two
vs. magnetic field at frequencies of 31 MHz under various
temperatures in HoB2C2. Magnetic fields up to 8 T were successive transitions of III- III”001 and III”001-II, indi-
applied along the[100]-axis. cated by arrows, are a common feature in the measure-
ments performed at 0.55, 1.4, 2.5, and 3.5 K. At 4.5K,
a re-entrant process of III-I-II phase transition was ob-
served between 2 T to 2.7 T. At 5.5 K, the I-IV phase
boundaryshowsabroadplateauaround1.4T.At6.5K,
symmetry strain ε (or ε ) corresponding to one com-
zx yz C showsamonotonousincreaseinphaseI.Itshouldbe
66
ponent in E(Γ34) doublet. The magnetic field depen- noted that the tetra-critical point exists at T∗ ∼ 4.5 K
dence of C44 in Fig. 5 shows two anomalies, a kink at andin a field H∗ ∼3 T appliedalong the principal[001]
1.8 T corresponding to the I-III phase transition and a
direction. This point is argued again in the magnetic
step anomaly at 0.2 T. The later transition shows hys-
phase diagram of Fig. 11.
tereticbehavior,whichmaybecausedbyadomaineffect
As shown in Fig. 2, the transverse C mode in zero
in phase III. 44
field shows considerable softening on the order of 20 %
with decreasing temperature. The softening of C is
In Fig. 6, we show the H-T phase diagramof HoB C 44
2 2
very much reduced in applied fields along the principal
which was obtained by ultrasonic and magnetization
[001]direction,as showninFig.8(a). The sharpminima
measurementsinfieldsappliedparalleltothebasalplane.
of C in fields below 3 T, shown by downward arrows,
Phase boundaries were determined by C vs. H (solid 44
66
indicate the IV-III transitionpoints. The I-IVtransition
triangles), C vs. T (grey circles), C vs. H (solid di-
66 44
points,whichhaveclearlybeenobservedintheresultsof
amonds) and C vs. T (solid reverse triangles). Open
44
C , werenotidentifiedinthe resultsofC inFig.8(a).
symbolsshowtheresultsofmagnetizationmeasurements 66 44
by Onodera et al.8 In the phase diagram of Fig. 6(a) we The shallow minima of C44 above 4 T up to 11 T corre-
spondtothetransitionfromtheparamagneticphaseIto
usethepreviousultrasonicresultsfromRef. 29. Inorder
the AFQ phase II. In Fig. 8(b), we show magnetic field
to verify the anisotropy of the AFQ phase II, the mag-
dependence of ∆C /C at various temperatures. The
netic fields wereappliedalongthe intermediatedirection 44 44
results of 2.2, 3.0, and 4.0 K show the III-II transition
with an angle θ = 22.5 deg. between the [100] and the
around 4 T, indicated by arrows. As shown in the inset
[110]. Furtherdetailsoftheultrasonicresultsareskipped
of Fig. 8(b), a hysteresis effect has been observed which
in the present paper for convenience.
suggest the first order class of the III-II transition. As
As one can see in Fig. 6, the AFQ phase II shows will be shown in the phase diagram of Fig. 11, phase IV
anisotropic behavior as a function of the field direction changes to the phase I with increasing fields at 5.2 K.
in the basal x-y plane. The upper critical magnetic field The magnetic field dependence of ∆C /C up to 11
44 44
H =3.9ToftheII-ItransitionforH k[110]shrinks T with H k [001] in Fig. 8(b) revealed the field induced
C[110]
to H = 2.0 T for H k [100], while the AFM phase I-II transition. In order to examine the II-I phase tran-
C[100]
III andthe phase IV behave mostly in an isotropic man- sitions in higher fields with H k [001], we have pursued
ner being independent of the field direction in the basal ultrasonicmeasurementsofC usingthe hybridmagnet
11
plane. The order parameter of phase II has stability (GAMA)upto30TinTsukubaMagnetLaboratory. We
against field H k [110] more than H k [100]. Actually, chose the longitudinal C mode because of its definite
11
phase II is stable only in the vicinity of the phase III-I ultrasonicecho signalas comparedto the relativelyfaint
boundary along H k [100] in Fig. 6(a). echosignalintransverseultrasonicmodes. Figure 9 rep-
5
FIG. 6: H-T phase diagrams of HoB2C2 with the fields applied along the (a) [100], (b) θ=22.5 deg. and (c) [110] directions.
DatapointsweredeterminedfromtheelasticanomaliesinC66 vs. H (solidtriangles),C66 vs. T (greycircles),C44 vs. T (solid
reverse triangles) and C44 vs. H (solid diamonds) as shown in Figs. 3- 5. Solid and dotted lines are guide for eyes. Previous
data of magnetization measured byOnodera et al. are also shown for comparison.
FIG. 8: Relative change of the elastic constant ∆C44/C44 of
the transverse modes at frequencies of 31 MHz under vari-
FIG. 7: Relative change of the elastic constant ∆C66/C66 of ous fields and temperatures in HoB2C2. Fields up to 11 T
thetransversemodesatfrequenciesof 31MHzundervarious
were applied along the [001]-axis. (a) shows temperature de-
fieldsandtemperaturesinHoB2C2. Fieldsupto8Twereap- pendence, (b) shows magnetic field dependence. Inset of (b)
pliedalongthe[001]-axis. (a)showstemperaturedependence,
shows hysteretic behavior around 3.5 K.
(b) shows field dependence.
II closes in the vicinity of critical field H =26.3 T.
C[001]
resents the magnetic field dependence of ∆C /C at
11 11 Figure 11 represents the magnetic phase diagram in
1.5 K for H k [001]from 5 T to 30 T. A sharp minimum
field along the principal [001]-axis up to 30 T. The gray
at 26.3 T has been found. The behaviors in AFQ phase
circles, solid squares, solid triangles, solid reversed tri-
transition of C are very similar to the ones of C for
11 66 angles and solid diamonds are the present results of C
66
H k [100] or [110].
andC . The open triangles and openreversedtriangles
44
Figure 10 represents the temperature dependence of representthetransitionsinthe presentresultsofC ob-
11
∆C /C under fields of 15, 17, 18, 20 and 30 T. The tained by using the hybrid magnet. There are three or-
11 11
C shows minima corresponding to the I-II phase tran- dered phases, the AFQ phase II, AFM phase III, and
11
sition, indicated by arrows,around 4 K and below 20 T. phaseIV inadditionto the paramagneticphaseI. Inthe
The minima shift slightly to lowertemperatureswith in- phase III, sub-phase III” is exist between 2 T and 4
001
creasingfields. Noanomalyintheresultat30Tindicates T.The magnetic neutronscatteringin sub-phaseIII” on
the absenceof the I-II transition. The distinct difference HoB C has not been reported yet. On the analogy of a
2 2
betweentheresultsof20Tand30Tsuggeststhatphase similarsub-phaseIII’onDyB C ,themagneticstructure
2 2
6
FIG.9: Relativechangeoftheelasticconstant∆C11/C11ata
fixedtemperatureof1.5Kdisplayedasafunctionofmagnetic
fieldalongthe[001]-axisupto30T.Measurementfrequencies
are 52 MHz.
FIG.11: MagneticH-T phasediagramsofHoB2C2withfields
applied along the [001]-axis. Data points were determined
fromtheelasticanomalies inC66 vs. T (greycircles), C66 vs.
H (solid triangles), C44 vs. H (solid reverse triangles), C44
vs. T (solid diamonds), C11 vs. T (open reverse triangles)
and C11 vs. H (open triangles) as shown in Figs. 7- 9. Inset
showsexpandedviewoftetra-criticalpoint. Solidanddotted
lines are guide for eyes.
C andC . Thereisadifferencebetweenthedata-plots
44 11
C and C that may be due to the mode difference or
66 44
samplesetting. Weuse the dataofC toobtainthe I-II
44
transition in high field. The point at 26.3 T obtained
by C in Fig. 9 suggests an upper phase boundary of
11
theAFQphaseIIinfields. Theclosedmagneticdiagram
FIG. 10: Relative change of the elastic constant ∆C11/C11
of the AFQ phase in the present HoB C resembles the
as a function of temperature at frequencies of 52 MHz under 2 2
various field applied along the[001]-axis. closed behavior in the AFQ phase II in Ce0.5La0.5B6.38
of sub-phase III” for H k[110]on HoB C is expected IV. CONCLUDING REMARKS
110 2 2
the altered form of AFM structure in phase III, which
themagneticmomentsarepartlyrotatedtotheadvanta- We have measured the elastic constants C , (C −
11 11
geous direction for the external magnetic field.25 On the C )/2, C , and C of HoB C . The softening of these
12 44 66 2 2
other hand, the boundary of phase III”001 for H k[001] modes is due to the pseudo triplet ground state of the
hasnotbeenobservedbymagnetizationmeasurementon system. The minima or kinks of C , C and C were
11 44 66
HoB2C2.8 Therefore,theoriginofphaseIII”001 wouldbe useful in particular to determine the transition points
a different from phase III” . in fields. We have obtained the H-T phase diagrams of
110
As one can see in inset of Fig. 11 , the four phases HoB C forthefieldsalongtheprincipal[001]axisandin
2 2
∗
meet each other at the tetra-critical point T ∼ 4.5 K the basal x-y plane. In the H-T phase diagrams, there
and H∗ ∼ 3.0 T. It is notable that the I-II and IV-III exists a tetra-critical point at T∗ ∼ 5.5 K, H∗ ∼ 0.75
phase boundaries approach the tetra-critical point tan- T for the basal plane and T∗ ∼ 4.5 K, H∗ ∼ 3.0 T for
gentialtoeachother,duetotheinterplayofseveralorder the principal [001]-axis,where two different interactions,
parameters. Similar multi-critical phenomena appear on magnetic dipole and electric quadrupole, are competing
theH-T phasediagramofanisotropicAFMsystems.36,37 with each other.
ThehysteresiseffectacrosstheII-IIIphaseboundaryen- The AFQ phase II is stable even in fields of H =
C[001]
sures the first order transition, and the discontinuity of 26.3 T along the [001] axis at absolute zero, while the
the elastic constants at the IV-III transition points may phase boundary shrinks to be H = 3.8 T for fields
C[110]
also indicate the first order transition. along the [110]-axis and H = 2.0 T for [100]-axis.
C[100]
The vertical I-II phase boundary from 4 T to 20 T is ThisanisotropyinthecriticalfieldsH ≫H >
C[001] C[110]
determinedbytheminimaintemperaturedependenceof H inHoB C isdominatedbytheanisotropyofthe
C[100] 2 2
7
quadrupolar RKKY-type quadrupole inter-site interac- H k [110] and O +O +O for H k [111] in cubic
yz zx xy
tion mediated by the conduction electrons in the tetrag- symmetry O .
h
onal system. The de Haas-van Alphen measurements In the case of tetragonal HoB C , when a field is ap-
2 2
by Watanuki et al. revealed the main Fermi-surface to pliedalongthe [001]-axis,the localsymmetryofthe rare
have a columnar shape indicating two-dimensional char- earth ion of C is lowered to C symmetry, while fields
4h 4
acter of the system.39 The band calculation of LaB2C2 applied [100]- or [110]-axes change C4h to C2. Since
also shows the two-dimensionalproperties, reflecting the the [001]-axis of Γ -type quadrupole O and/or O2
2 xy 2
strong hybridization of the 5d orbitals of La with the 2p has a highest symmetry of three principal [100]-, [110]-
orbitals in B-C sheets.40 The anisotropic band structure and [001]-axis,this causes no modulation in the Γ -type
2
willplayaroleintheanisotropicquadrupoleinter-sitein- quadrupole when H k [001]. As a consequence, the Γ -
2
teraction,whichbringsabouttheanisotropicbehaviorof type AFQ order parameter O and/or O2 would be
xy 2
the AFQ phase of HoB2C2 and DyB2C2 under magnetic stuck on the basal plane even in magnetic field parallel
fields. tothebasalplaneduetothetetragonalityofthesystem.
It is useful to demonstratethe symmetry propertiesof
the quadrupole moments under the applied fields along
thehighsymmetry[100]-,[110]-and[001]-axis. Onemay
Acknowledgments
applied the symmetry argument of AFQ order param-
eters in cubic CeB , which was proposed by Shiina et
6
al.41,42,43,tothe presenttetragonalHoB C system. For The authors would like to thank J. Igarashi, O.
2 2
a finite magnetic field, the symmetry of the system is Sakai, R. Shiina, and H. Onodera for fruitful discus-
loweredtokeepthefieldinduceddipolemoment,namely sions. The present work was supported by a Grant-in-
angularmomentumJ ,J andJ to be invariant. Inthe Aid for Scientific Research Priority Area ”Skutterudite”
x y z
case of CeB , due to the Γ -type AFQ order parame- (No. 15072206) of the Ministry of Education, Culture,
6 5
ter,linearcombinationsofquadrupolemomentsresultin Sports,ScienceandTechnology. T.Yanagisawawassup-
cubic symmetry. In an applied magnetic field, the high- ported by Research Fellowships of the Japan Society for
est symmetry axis for these moments are related to field the Promotionof Science for Young Scientists in present
directions as follows; O for H k [001], O +O for research.
xy yz zx
∗ Also at Institute of Pure and Applied Physical Sciences, 12 H. Yamauchi, H. Onodera, M. Ohashi, K. Ohoyama, T.
University of California at San Diego, La Jolla, California Onimaru, M. Kosaka, M. Ohashi, and Y. Yamaguchi, J.
92093, USA. Phys. Chem. Solids 60, 1217 (1999).
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