広島大学学術情報リポジトリHiroshima University Institutional Repository

TitleChiral Soliton Lattice Formation in Monoaxial HelimagnetYb(Ni1-xCux)3Al9

Auther(s)Matsumura, Takeshi; Kita, Yosuke; Kubo, Koya; Yoshikawa,Yugo; Michimura, Shinji; Inami, Toshiya; Kousaka, Yusuke;Inoue, Katsuya; Ohara, Shigeo

CitationJournal of the Physical Society of Japan , 86 (12) :124702-1 - 124702-12

Issue Date2017-11-01

DOI10.7566/JPSJ.86.124702

Self DOI

URLhttp://ir.lib.hiroshima-u.ac.jp/00046500

RightCopyright (c) 2017 The Physical Society of JapanCopyright (c) 2017 一般社団法人日本物理学会This is not the published version. Please cite only thepublished version. この論文は出版社版でありません。引用の際には出版社版をご確認ご利用ください。

Relation

Journal of the Physical Society of Japan

Chiral Soliton Lattice Formation in Monoaxial Helimagnet Yb(Ni1−xCux)3Al9

Takeshi Matsumura1,2,6,7, Yosuke Kita1, Koya Kubo1, Yugo Yoshikawa1, Shinji Michimura3, ToshiyaInami4, Yusuke Kousaka5,6, Katsuya Inoue5,6, and Shigeo Ohara8

1Department of Quantum Matter, AdSM, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8530, Japan2Institute for Advanced Materials Research, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8530, Japan

3Department of Physics, Faculty of Science, Saitama University, Saitama 338-8570, Japan4Synchrotron Radiation Research Center, National Institutes for Quantum and Radiological Science and Technology,

Sayo, Hyogo 679-5148, Japan5Department of Chemistry, Graduate School of Science, Hiroshima University, Higashi-Hiroshima, Hiroshima

739-8526, Japan6Chirality Research Center (CResCent), Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan

7Center for Emergent Condensed Matter Physics (ECMP), Hiroshima University, Higashi-Hiroshima, Hiroshima739-8526, Japan

8Department of Physical Science and Engineering, Graduate School of Engineering, Nagoya Institute of Technology,Nagoya 466-8555, Japan

Helical magnetic structures and their responses to external magnetic fields in Yb(Ni1−xCux)3Al9 with a chiral crystalstructure of the space group R32 have been investigated by resonant X-ray diffraction. It is shown that the crystal chi-rality is reflected in the helicity of the magnetic structure by a one-to-one relationship, indicating that an antisymmetricexchange interaction mediated via the conduction electrons exists. When a magnetic field is applied perpendicular to thehelical axis (c axis), the second-harmonic peak of (0, 0, 2q) develops with increasing field. The third-harmonic peak of(0, 0, 3q) has also been observed for the x=0.06 sample. This result provides strong evidence for the formation of a chiralmagnetic soliton lattice state, a periodic array of chiral twist of spins, which has been suggested by the characteristicmagnetization curve. The helical ordering of magnetic octupole moments accompanying the magnetic dipole order hasalso been detected.

Journal Ref: J. Phys. Soc. Jpn., 86, 124702 (2017).http://dx.doi.org/10.7566/JPSJ.86.124702

1. IntroductionChirality is one of the most fundamental elements of sym-

metry in nature. It plays an important role in various phenom-ena ranging from biological functions to the physical proper-ties of inorganic substances.1) In magnetic materials lackingthe local inversion center for the two-ion exchange interac-tion, a spiral magnetic order is often stabilized by the antisym-metric Dzyaloshinskii–Moriya (DM) interaction, giving riseto distinct physical properties.2, 3) The simultaneous appear-ance of electric polarization with the spiral magnetic orderis a typical manifestation of such effects.4) In chiral magneticmaterials without both inversion and mirror symmetries, a he-lical magnetic order with a fixed sense of spin rotation can bestabilized. When a magnetic field is applied to such a system,a characteristic arrangement of the topological spin structureis often stabilized through competition between the Zeemanenergy and the twisting force from the DM interaction. In cu-bic B20-type compounds such as MnSi with the space groupP213, for example, the helical spin structure transforms into ahexagonal lattice condensate of magnetic skyrmions.5–7) Theskyrmion state is a long-range pattern of twisted spin arrange-ments realized in magnetic fields, which become more stablein two-dimensional configurations in thin films.8, 9)

In monoaxial chiral helimagnets such as CrNb3S6 (spacegroup P6322), when a magnetic field is applied perpendic-ular to the helical axis, the helical ground state transformsinto a periodic array of incommensurate chiral spin twists,which separates the ferromagnetically aligned commensurate

region.10–14) This is a nonlinear order of topological spinstructure and is called a chiral soliton lattice (CSL). Thesematerials are expected to provide a new functionality that isoperated by tuning the number of skyrmions or solitons in thesample.15, 16)

In the present paper, we report on the new monoaxial chi-ral helimagnet system of Yb(Ni1−xCux)3Al9, in which theCSL state is expected to be realized. The rare-earth com-pound YbNi3Al9 has a chiral crystal structure with the spacegroup R32 (No. 155), which lacks both space inversion andmirror symmetries.17, 18) The main block of the crystal struc-ture is shown in Fig. 1 (drawn with VESTA19)). The physicalproperties of YbNi3Al9 have been studied as those of an Yb-based heavy-fermion compound.20–25) Yb ions form a two-dimensional honeycomb lattice in the c plane, which is sep-arated by c/3=9.121 Å from the neighbouring Yb layer byfive Al and two Ni layers. Since this is much larger than thenearest-neighbor distance of a/

√3=4.199 Å within a layer,

the relation between the two-dimensionality and the heavy-fermion state has also been of interest. The detailed study ofthis compound from the viewpoint of its chirality started withthe discovery of a characteristic magnetization process that isreminiscent of a CSL state.26) By substituting Ni with Cu, itwas discovered that the M(H) curve behaves similarly to thatof CrNb3S6,27) in which the CSL state has clearly been iden-tified.12–14)

The exchange interaction in metallic YbNi3Al9 is consid-ered to be of the Ruderman–Kittel–Kasuya–Yosida (RKKY)

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J. Phys. Soc. Jpn.

Fig. 1. (Color online) Crystal structure of YbNi3Al9 with the space groupR32.17, 18) We call the structure with x=0.3332, y=0.0056, z=0.08517 for the18 f site of Ni the right (R) crystal and its mirror-reflected structure the left(L) crystal. Only the Yb2(6c)+Al3(9e)+2Ni3(18 f )+2Al3(18 f ) block layer atthe bottom of the unit cell is shown. This block is shifted by (2/3, 1/3, 1/3)and (1/3, 2/3, 2/3). The other Al layers at the 6c and 9d sites are omitted.VESTA was used to draw the figure.19)

type. Since the crystal is chiral, there must be some antisym-metric contribution to the RKKY mechanism in the form ofDi j · Si × S j, which is generally called the DM interaction.2, 3)

Since the microscopic mechanism given by Moriya is basedon the superexchange interaction in insulators, the mechanismof the RKKY-type DM interaction is an important subject tobe studied. The question is how the crystal chirality is trans-ferred to the spin system of f electrons and conduction elec-trons.

YbNi3Al9 is a metallic compound which orders at TN=3.5K. It is reported that the magnetic structure of YbNi3Al9 ishelical with propagation vector (0, 0,∼0.8) and the momentslying in the c plane. By applying a magnetic field perpendicu-lar to the c axis, the helimagnetic order jumps to a ferromag-netic state at a critical field of Hc=1 kOe.22, 23) By substitutingNi with Cu, both TN and Hc increase. An anomalous M(H)curve reminiscent of a CSL state is observed for x=0.06.26)

However, the relationship between the crystal chirality andthe magnetic helicity, the Cu concentration dependence of thehelimagnetic structure, and detailed dependences on the tem-perature and magnetic field have not yet been studied. Theaim of the present study is to clarify these properties experi-mentally by using resonant X-ray diffraction (RXD).

Experimentally, how to observe a chiral state has long beena subject of interest. Recent progress in the transmission elec-tron microscopy method has made it possible to capture real-space images of skyrmions and CSLs.5–8, 12) It is also of fun-damental importance to capture the spatially averaged struc-ture as a Fourier transform of a real-space image by neutronand X-ray diffraction methods. Polarized neutron diffractionis a powerful method for observing the helical magnetic struc-ture and for determining the helicity.28, 29) One drawback ofthis method is that it is difficult to perform an experimentin magnetic fields because the neutron spin state is affectedby the magnetic field. X-ray diffraction, on the other hand,can be used both at zero field and in magnetic fields with-out any differences. The helicity of a magnetic spiral can bestudied by using circularly polarized X-rays.30–32) By utiliz-ing resonance at an absorption edge of the magnetic element,the scattering cross section is enhanced, making it possibleto detect signals from an ordered structure more efficiently.33)

The magnetic skyrmion state in a chiral magnet has been de-tected by RXD.34) Furthermore, resonant scattering has sensi-tivity to higher-order anisotropy (multipole moments) of bothmagnetic and nonmagnetic nature.35–37) This sensitivity can

45¡

q PR

HRCP LCP

sp

k'

k

Z

X

Y

q

q

Sample

Detector

[0 1 0]

[0 0 1]

e

PhaseRetarder

-1

0

1

P2, P

3

-0.1 0 0.1Dq PR (deg.)

LCP RCP

P2

P3

Fig. 2. (Color online) Scattering configuration of the experiment with aphase retarder system inserted in the incident beam. The inset figure showsthe ∆θPR = θPR − θB dependence of the polarization state using the Stokesparameters P2 and P3. The vertical dotted lines represent the positions of LCPand RCP states. The beam is depolarized in the region around ∆θPR ≈ 0.

sometimes be applied to determine crystal chirality by usinga circularly polarized beam.38–40)

This paper is organized as follows. In Sect. 2, the experi-mental procedure is described, including the details of the cir-cularly polarized X-ray beam. The experimental results andthe analyses are described in Sect. 3. First, in Sect. 3.1, theone-to-one relationship between the crystal chirality and thehelimagnetic structure is described. Comparison of the exper-imental results with the helimagnetic structure model is per-formed in Sect. 3.2. In Sects. 3.3 and 3.4, the temperatureand magnetic field dependences of the helimagnetic order arepresented. The resonant nature of the signal is described inSect. 3.5. We show that the E2 resonance involves a signalfrom a magnetic octupole, which accompanies the helical or-der of the magnetic dipole. In Sect. 4, we discuss the originof the octupole moment, the temperature-dependent pitch ofthe helical structure, and the possibility of the CSL state inYb(Ni1−xCux)3Al9. The present study is summarized in Sect.5.

2. ExperimentSingle crystals of Yb(Ni1−xCux)3Al9 were prepared by an

Al-flux method following the procedure described in the liter-ature.26) The starting composition x′ of Cu sealed in a quartzampoule was set to five times the target composition x. Theactual Cu concentration x in the obtained crystal was verifiedby electron probe microanalysis and was confirmed to followthe relation x ∼ 0.2x′, as reported previously.26) We also ver-ified the sample quality by magnetic susceptibility, magneti-zation, and electrical resistivity measurements and obtainedconsistent results with those reported in the literature.22, 26)

RXD experiments were performed at BL22XU at SPring-8. The c-plane surfaces of the samples were mirror-polished,and the samples were mounted in a vertical-field 8 T super-conducting cryomagnet equipped with a 3He cryostat insertso that the c axis was perpendicular to the magnetic field andcoincided with the scattering vector k′ − k ∥ Z. The scatteringgeometry is shown in Fig. 2. The incident X-ray energy wastuned to the L3 edge of Yb.

We used a diamond phase retarder system to tune the hor-izontally polarized incident beam to a circularly polarized

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J. Phys. Soc. Jpn.

state.41) By rotating the angle of the diamond phase plate θPRabout the 220 Bragg angle θB, where the scattering plane istilted by 45, a phase difference arises between the σ and πcomponents of the transmitted beam. The phase difference isapproximately proportional to 1/(θPR − θB). This allows usto tune the incident linear polarization to right-handed circu-lar polarization (RCP) and left-handed circular polarization(LCP) by changing ∆θPR = θPR − θB. The polarization stateof the incident beam as a function of ∆θPR is shown in Fig. 2using the Stokes parameters P2 (+1 for RCP and −1 for LCP)and P3 (+1 for σ and −1 for π linear polarization).42) In thehorizontal scattering plane configuration in the present exper-iment, the incident linear polarization is π when ∆θPR is large.P1 (+1 for 45 and −1 for −45 linear polarization) is zero inthe present setup.

We define RCP as εR = (εσ + iεπ)ei(k·r−ωt) and LCP asεL = (εσ − iεπ)ei(k·r−ωt). We verified the helicity of the in-cident photon after transmitting the phase retarder by measur-ing the resonant scattering intensity of a forbidden reflectionfrom P6122-type CsCuCl3, where the intensity ratio betweenRCP and LCP X-rays depends on the reflection index and isexactly determined by the P6122 space group.38)

The crystal chirality of the sample was determined by usinga laboratory-based X-ray diffraction system (Bruker APEX-II) and it was also verified in the RXD experiment at thebeamline. In the former method, using a Mo Kα X-ray beam,the Flack parameter was deduced by analyzing the intensi-ties of many reflections, which resulted in either 0 (R) or 1(L). Our definition of the crystal chirality is shown in Fig. 1.Then, R and L samples were selected for each Cu concentra-tion for use with the RXD experiment. At the beamline, theenergy dependences of the (1, 1, 24) and (1, 1, 24) fundamen-tal Bragg peak intensities were measured around the absorp-tion edge of Yb, which is shown in Fig. A·1 in Appendix A.The spectrum exhibited a contrasting energy dependence atthe edge depending on the chirality of the crystal. It was con-sistent with the calculated spectrum assuming the predeter-mined crystal chirality, confirming that the irradiated spot ofthe sample in the RXD experiment has exactly the same chi-rality as that determined in the laboratory X-ray diffraction. Itwas also confirmed at the beamline that the (1, 0, 3n + 1) re-flections are allowed and the (1, 0, 3n) and (1, 0, 3n−1) reflec-tions are forbidden as shown in Fig. A·2. This fact guaranteesthe threefold symmetry of the sample about the c axis. If theforbidden reflections were observed, it would have meant thatthe [1 1 0] axis was mixed with the a axis due to the stackingfault of 60.

3. Results and Analysis3.1 Crystal chirality and the helical magnetic structure

First, we demonstrate that the crystal chirality and the he-licity of the helical magnetic structure have a one-to-one rela-tionship. A typical example is shown in Fig. 3 for the R andL crystals with x=0.06. The reciprocal scans along (0, 0, L)around the (0, 0, 27) fundamental Bragg peak using RCP andLCP X-rays exhibit opposite behavior for the R and L crystalsas well as those for the (0, 0, 27−q) and (0, 0, 27+q) peaks. Inthe R(L) crystal at (0, 0, 27− q), the intensity is strong (weak)for LCP and weak (strong) for RCP. This relation is reversedat (0, 0, 27+q). The X-ray energy of 8.944 keV corresponds tothe E1 (2p ↔ 5d) resonance. The energy dependence of the

27.4527.4L (r.l.u.)

RCP LCP

(0, 0, 27 + q)

E = 8.944 keV (E1) T = 1.5 K

x = 0.06 (R)

200

150

100

50

0

Inte

nsity

(cn

ts/s

)

26.626.55L (r.l.u.)

RCP LCP

(0, 0, 27 Ð q)

Yb(Ni1-xCux)3Al9

x = 0.06 (R)

400

300

200

100

0

Inte

nsity

(cn

ts/s

)

26.626.5526.5L (r.l.u.)

RCP LCP

(0, 0, 27 Ð q)

x = 0.06 (L)

27.527.4527.4L (r.l.u.)

RCP LCP

(0, 0, 27 + q)

x = 0.06 (L)

Fig. 3. (Color online) Reciprocal space scan along (0, 0, L) around the(0, 0, 27) fundamental peak for the x=0.06 sample with right and left chiralityusing RCP and LCP X-rays. Solid lines are the fits with Gaussian functions.

intensity demonstrating the resonance feature will be shownlater. In Fig. 3, the peaks are observed at an incommensuratewave vector of q=0.445 for the L crystal and q=0.425 for theR crystal. This difference in the q value is due to the differ-ence in the sample quality, i.e., a slight difference in the Cuconcentration, and is unrelated to the chirality.

More detailed information can be extracted from the ∆θPRscans, which are shown in Fig. 4. This figure shows notonly the intensity relations for the RCP and LCP X-rays at(0, 0, 27 ± q) but also the overall feature of the incident polar-ization dependence as a function of ∆θPR. The incident polar-ization state varies with ∆θPR as shown in the inset of Fig. 2.We again observe from this measurement that the crystal chi-rality and the helicity of the magnetic structure have a one-to-one relationship. In addition, the relationship does not changewith the Cu concentration between x=0 and x=0.06. The solidlines in the figures represent the calculated curves expectedfrom the helical magnetic structure with the moments lyingin the c plane and propagating along the c axis. Next, we de-scribe the analysis of the above experimental results.

3.2 Magnetic structureThere are two Yb atoms for the 6c site of the R32 space

group: Yb-1 at d1 = (0, 0, z) and Yb-2 at d2 = (0, 0, z),where z=0.167∼ 1/6. In the present single-q magnetic struc-ture, the magnetic moments µ1, j and µ2, j of Yb-1 and Yb-2, respectively, on the jth lattice point at r j = (n1, n2, n3),(n1+2/3, n2+1/3, n3+1/3), and (n1+1/3, n2+2/3, n3+2/3),where n1, n2, and n3 are integers, are generally expressed as

µ1, j = m1eiq·r j + m∗1e−iq·r j , (1a)

µ2, j = m2eiq·r j + m∗2e−iq·r j , (1b)

where m1 and m2 are the magnetic amplitude vectorsof Yb-1 and Yb-2, respectively. In the present case ofYb(Ni1−xCux)3Al9, since the moments are expected to be or-dered within the c plane, m1 and m2 can generally be written

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J. Phys. Soc. Jpn.

Table I. Relation among the phase φ, Fourier component m, scatteringvector Q, magnetic structure factor Z(1)

dip, sign of C2, and crystal chirality.

φ m Q Z(1)dip C2 Crsytal

π/2 x + iy (0, 0, 3n ± q) (1,±i, 0) ± R−π/2 x − iy (0, 0, 3n ± q) (1,∓i, 0) ∓ L

2.0

1.5

1.0

0.5

0.0

Inte

nsity

(arb

. un

it)

(0, 0, 27 + q )(0, 0, 27 Ð q )

x = 0 (L)

Yb(Ni1-xCux)3Al9 E = 8.944 keV (E1)

(0, 0, 27 + q )(0, 0, 27 Ð q )

x = 0.06 (L)

2.0

1.5

1.0

0.5

0.0

Inte

nsity

(arb

. un

it)

-0.1 -0.05 0 0.05 0.1

Dq PR (deg.)

(0, 0, 27 + q )(0, 0, 27 Ð q )

x = 0 (R)

-0.1 -0.05 0 0.05 0.1

Dq PR (deg.)

(0, 0, 27 + q )(0, 0, 27 Ð q )

x = 0.06 (R)

Fig. 4. (Color online) Incident polarization (∆θPR) dependence of the in-tensities at (0, 0, 27 ± q) for x=0 and x=0.06 crystals with right and left chi-rality. The background has been subtracted. Solid lines are the calculationsdescribed in the text.

as m1 = m1(x+ eiφ y) and m2 = m2eiδ(x+ eiφ y), where x and yrepresent the unit vectors along the x and y axis, respectively,which are taken perpendicular to the c (z) axis. The irreduciblerepresentation of the m-vector for q = (0, 0, q), where q is anincommensurate value, is written as x± iy. Therefore, φ is ei-ther π/2 or −π/2. The phase difference between µ1, j and µ2, jis represented by δ. The above expression for the magneticstructure can be reduced to

µ1, j = m1x cos q · r j + y cos(q · r j + φ)

, (2a)

µ2, j = m2x cos(q · r j + δ) + y cos(q · r j + φ + δ)

. (2b)

Since φ = ±π/2, the above expression describes a perfect he-lical structure with helicity ±1 in which the adjacent Yb-1 (orYb-2) moments on the neighboring layers along the c axismake a fixed angle of 2πq/3. On the other hand, the angle be-tween the Yb-1 and Yb-2 moments within a layer, the formerat (0, 0, z) and the latter at (2/3, 1/3, 1/3) + (0, 0, z) in Fig. 1,is described by the parameter δ. This angle is reported to be20.5 for x=0.43) Note that our experimental results presentedin this paper, which were collected only along the (0, 0, L)line, are not sufficiently sensitive to determine the δ value.

The E1 resonant scattering amplitude from the magneticdipole order is proportional to i(ε′ × ε) · Z(1)

dip,33, 35) where

Z(1)dip =

∑j,d

µ j,de−iQ·(r j+d) (3)

Fig. 5. (Color online) Magnetic structure of Yb(Ni1−xCux)3Al9 forx=0.06. Only the moments on the Yb-1 sites are shown.

represents the magnetic structure factor at Q = k′ − k. AtQ = (0, 0, 3n ± q), Z(1)

dip = (1,±i, 0) when φ = π/2, and Z(1)dip =

(1,∓i, 0) when φ = −π/2. The scattering amplitude matrix,as defined in Appendix B, for the magnetic structure factor(1,±i, 0) is expressed as

FE1 =

(0 i cos θ

−i cos θ ∓ sin 2θ

). (4)

Then, following the method described in Appendix B, we cancalculate the coefficients Cn (n=0–3) defined in Eq. (B·5),which express the scattering cross sections. The parametersC2/C0 and C3/C0, as normalized by the total cross sectionC0, correspond to the intensity terms proportional to P2 andP3, respectively. When C2 > 0, the intensity is stronger forthe RCP (∆θPR > 0). For Q = (0, 0, 27 ± q), when φ = π/2,C2/C0 = ±0.71 and C3/C0 = −0.47 are obtained. The ob-served intensity is a superposition of the P2 and P3 terms.The experimental data in Fig. 4 show that at Q = (0, 0, 27+q)C2 > 0 in the R-crystal and C2 < 0 in the L-crystal. Thismeans that φ = π/2 (m = x+iy) in the R-crystal and φ = −π/2(m = x − iy) in the L-crystal. These relations are summa-rized in Table I. The magnetic structure is shown in Fig. 5 forx=0.06. The Yb-2 moment is not shown because the relativeangle with the Yb-1 moment is unknown. The solid lines inFig. 4 are the calculations using the C2/C0 and C3/C0 val-ues calculated above, which agree well with the experimentaldata.

3.3 Temperature dependenceFigure 6 shows the (0, 0, L) peak profile for x=0 (L) and

x=0.06 (L) samples. With increasing T , the intensity de-creases and vanishes at TN, indicating that the resonant signalis of magnetic origin. In addition, the peak position shifts withthe temperature. It is also noteworthy that the direction of thepeak shift for x=0.06 is opposite to that for x=0. It dependson the Cu concentration x but does not depend on the chiralityof the crystal.

The parameters obtained from the (0, 0, L) scans are sum-marized in Fig. 7. For x=0, the q value decreases with de-creasing T and saturates at ∼ 0.818 at the lowest temperature,indicating that the helical structure is incommensurate withthe lattice. The T dependence of the q value becomes weakfor x=0.02. Surprisingly, at x=0.04, the direction of the T de-pendence is reversed and the q value increases with decreas-

4

J. Phys. Soc. Jpn.

400

300

200

100

0

Inte

nsity

(cn

ts/s

)

26.5826.5626.54

L (r.l.u.)

(0, 0, 27 – q ) RCP

x = 0.06 (L) 1.5 K 2.5 K 3.5 K 4.5 K 5.5 K 6.0 K 6.5 K

27.4627.4427.42

L (r.l.u.)

(0, 0, 27 + q ) LCP

100

80

60

40

20

0

Inte

nsity

(cn

ts/s

)

26.2026.15

L (r.l.u.)

(0, 0, 27 – q )RCP

0.9 K 1.8 K 2.2 K 2.6 K 3.0 K 3.4 K

Yb(Ni1-xCux)3Al9

x = 0 (L)

27.8527.80

L (r.l.u.)

(0, 0, 27 + q ) LCP

E = 8.944 keV (E1)

Fig. 6. (Color online) Temperature dependence of the peak profile for x=0(L) and x=0.06 (L).

0.830

0.825

0.820

0.815

x = 0

Yb(Ni1-xCux)3Al9

(a)

0.620

0.615

x = 0.02(b)

0.555

0.550q (

r.l.u

.)

x = 0.04(c)

0.445

0.440

0.435

0.430

0.4256420

T (K)

x = 0.06

(d)

1.0

0.5

0Inte

grat

ed In

tens

ity (

arb.

uni

t)

6420T (K)

x =00.020.040.06

(e)

0.8

0.6

0.4

q (

r.l.u

.)

6420x (%)

70

60

50

40

30

L 0 (

)

(f)

Fig. 7. (Color online) (a)–(d) Temperature dependence of the q value of thepropagation vector. The vertical dashed line represents TN. (e) Temperaturedependence of the normalized intensity. (f) Cu concentration dependence ofthe q value (closed circles) and the pitch L0 = c/q (open circles) at the lowesttemperature.

ing T . At x=0.06, the T dependence becomes strong again. Itseems that the direction of the shift in the q value is reversedat around x=0.03, where q ∼ 0.6.

Figure 7(e) shows the T dependence of the integrated inten-sity for all the Cu concentrations. TN increases roughly pro-portionally to x, which is consistent with the literature.26, 44)

800

600

400

200

0

Inte

nsity

(cn

ts/s

)

20.6520.6020.55

L (r.l.u.)

04 6 8 9

9.510 kG // b*

7 8.5

Yb(Ni1-xCux)3Al9 x = 0.06 (L)

(a)

E = 8.934 keV (E2)

(0, 0, 21 Ð q ) T = 1.6 KRCP

80

60

40

20

0

Inte

nsity

(cn

ts/s

)

20.320.220.1

L (r.l.u.)

46

78

8.5

99.5 10 kG

0

(b) (0, 0, 21 Ð 2q )

Fig. 8. (Color online) Magnetic field dependence of the peak profile forx=0.06 measured with the RCP photon at the E2 resonance energy. (a) Firstharmonic peak at (0, 0, 21 − q). (b) Second harmonic peak at (0, 0, 21 − 2q).The field is applied perpendicular to the c axis (∥ b∗).

In addition, the shift of the q value mentioned above, regard-less of its direction, is almost proportional to the intensity,which reflects the development of the ordered moment. Thissuggests that the q-shift is associated with the magnitude ofthe ordered moment. In Fig. 7(f), we show the x dependenceof the q value and the pitch L0 = c/q at the lowest temper-ature. L0 should be compared with the interlayer distance ofc/3 ∼ 9.1 Å. The angle between the moments on neighbor-ing layers is calculated as 2πq/3, i.e., 98.2, 74.0, 66.4, and53.4 for x=0, 0.02, 0.04, and 0.06, respectively, at the lowesttemperature.

3.4 Magnetic field dependenceFigure 8(a) shows the magnetic field dependence of the

peak profile of (0, 0, 21 − q) for the x=0.06 (L) sample, mea-sured at the E2 (2p ↔ 4 f ) resonance energy of 8.934 keV,where the signal-to-noise ratio was much higher than that atthe E1 resonance. The energy dependence of the resonant sig-nal will be shown later. We used the RCP photon with strongerscattering intensity for the L-crystal than the LCP photon. Theresult for the second-harmonic peak measured at (0, 0, 21−2q)is shown in Fig. 8(b). With increasing H, the peak positionshifts to the fundamental Bragg peak at (0, 0, 21) and theintensity of the first-harmonic (q) peak gradually decreases.On the other hand, the intensity of the second-harmonic (2q)peak, which does not exist at zero field, gradually increaseswith increasing H.

The magnetic field dependence of the q values and the in-tegrated intensities of the q and 2q peaks for x=0.02, 0.04,and 0.06 are summarized in Fig. 9. Although all these mea-surements were performed using the L-crystals, the results donot depend on the crystal chirality. For all the Cu concentra-tions, it is commonly observed that the q value decreases withincreasing H, which becomes more rapid at higher fields onapproaching the critical field Hc. Note that q does not decrease

5

J. Phys. Soc. Jpn.

0.62

0.60

x = 0.02 1.5 K

Yb(Ni1-xCux)3Al9

0.54

0.52

x = 0.04 1.5 K

1.0

0.5

0.0

x = 0.02q

2q (x 4)

1.0

0.5

0.0

Inte

gra

ted I

nte

nsity

(arb

.un

it)

x = 0.04

2q (x 4)

q

1.0

0.5

0.01050

Magnetic Field (kG)

q

2q (x 4)

x = 0.06

0.44

0.42

0.40

0.38

0.36

q (r

.l.u

.)

1050

Magnetic Field (kG)

x = 0.06 1.6 K

0.375

Fig. 9. (Color online) Magnetic field dependence of the q values (left col-umn) and the integrated intensities (right column). The integrated intensitiesfor the 2q peak are multiplied by four.

continuously to zero but jumps to zero at Hc. The integratedintensity of the q peak gradually decreases with increasing Hand also jumps to zero at Hc. These results show that the tran-sition at Hc is of first-order. It is also a common characteristicthat the 2q peak is gradually induced with increasing H. Thisis observed even for the x=0.02 sample with µ0Hc=3 kG.

Another noteworthy result is the locking-in behavior of thehelical magnetic propagation vector at q=0.375=3/8 between9 and 9.5 kG for x=0.06. As observed in Fig. 8, the peak po-sition does not change in this field region. The 2q peak is alsofixed at 2q=3/4. This result shows that some coupling existsbetween the helimagnetic structure and the lattice, althoughthe magnetic anisotropy in the c plane is considered to be verysmall. There is no hysteresis in this behavior, which can beobserved in both field-increasing and -decreasing processes.It is also interesting that the lock-in does not seem to exist atq=3/7=0.4286 at around 5 kG. This suggests that the lock-inis more associated with the 2q peak, which develops at highfields, than the q peak existing from zero field to high fields.

At 9.5 kG, slightly below the critical field of 10 kG for theferromagnetic state in the x=0.06 (L) sample, we searched formore higher-order harmonics and successfully detected thethird-harmonic peak. The result is shown in Fig. 10. For allthe harmonics, the scattering intensity for the RCP photon isstronger than that for the LCP photon, indicating that the mag-netic helicity of the modulated structure giving the higher har-monic is the same as that of the original helical structure ofthe first harmonic. The fourth-harmonic peak was too weak tobe recognized above the background. The integrated intensi-ties of the 2q and 3q peaks are approximately 6 and 50 timesweaker, respectively, than that of the q peak.

The H dependence of the q value directly shows that thepitch of the helimagnetic structure increases with increasingH. In addition, the appearance of the higher-harmonic peaksshows that some additional structure is superimposed with in-creasing H. It is noteworthy that the H dependence of the q

20.6420.60

L (r.l.u.)

(0, 0, 21 – q)

20.2720.23

L (r.l.u.)

(0, 0, 21 – 2q)

x4

400

300

200

100

0

Inte

nsity

(cn

ts/s

)

19.8919.85

L (r.l.u.)

(0, 0, 21 – 3q)

8.934 keV (E2)1.6 K9.5 kG // b*

x8

Yb(Ni1-xCux)3Al9 x = 0.06 (L)

RCPLCP

Fig. 10. (Color online) L-scan profiles for the higher-harmonic peaks of(0, 0, 21 − nq) (n = 1, 2, 3) observed at 9.5 kG for x=0.06 (L).

and 2q peak intensities shown in Fig. 9 is very similar to thecalculation for the chiral sine-Gordon model (see Fig. 25 ofRef. 10 or Fig. 13 of Ref. 11). This result, as well as the Hdependence of q which is associated with L0/LCSL (Fig. 13 ofRef. 10 or Fig. 38 of Ref. 11), strongly indicates that a CSLstate is formed in Yb(Ni1−xCux)3Al9, especially for x=0.06.

3.5 Energy spectrumFigure 11 shows the energy dependence of the (0, 0, 27−q)

peak at 0 kG and the (0, 0, 27 − 2q) peak at 9 kG measuredfor x=0.06 (L). The two resonant peaks are well separated at8.934 and 8.944 keV. The former can be assigned to the E2resonance (2p ↔ 4 f ) peak and the latter to the E1 resonance(2p ↔ 5d) peak. In the tail on the lower-energy side of the(0, 0, 27−q) peak we can see weak nonresonant magnetic scat-tering. The energy dependence for the second-harmonic peakalso exhibits resonances at the E2 and E1 energies. No non-resonant signal was observed. This shows that the resonantsignal of the second harmonic can be ascribed to the helicalmagnetic order itself and not to a possible lattice deformationinduced by the magnetic order. These features of the energydependence were commonly observed for other Cu concen-trations.

3.6 Contribution of magnetic octupole to the E2 resonanceFigure 12 shows the ∆θPR dependence of the (0, 0, 27 − q)

peak intensity at zero field and the (0, 0, 27 − 2q) peak inten-sity at 9.9 kG at the E2 resonance energy. We notice that the∆θPR dependence appears to be different from that in Fig. 4for the E1 resonance. The E2 intensity is strongly enhancedwhen ∆θPR is tuned to the LCP or RCP position. The resultthat the two ∆θPR dependences are very similar shows thatthe structure factors for the q and 2q peaks are almost thesame. By fitting these ∆θPR dependences using Eq. (B·5), weobtain three parameters, C0, C2, and C3. The parameters ob-tained for various Q vectors for the first harmonic peak from(0, 0, 6 ± q) to (0, 0, 30 ± q) for the x=0.06 (R) sample areshown in Fig. 13. The parameters C2 and C3 are normalizedby C0. Note that the sign of C2 in Fig. 13 is opposite to the onededuced from Fig. 12 because the crystal chirality is opposite.With respect to the E1 resonance, the Q-dependence data inFig. 13(a) can be well reproduced by the scattering amplitude

6

J. Phys. Soc. Jpn.

300

200

100

0

Inte

nsity

(cn

ts/s

)

8.968.948.92

X-ray Energy (keV)

x = 0.06 (L) T = 1.5 K

0 kG

(0, 0, 27 Ð q )(a)

8.968.948.92

X-ray Energy (keV)

1000

500

m (c

mÐ

1)

(0, 0, 27 Ð 2q )

9 kG // b*

(b)

RCP

Fig. 11. (Color online) Energy dependence of (a) the (0, 0, 27 − q) peakintensity at zero field and (b) the (0, 0, 27 − 2q) peak intensity at 9 kG forx=0.06 (L) measured with RCP incident photons after background subtrac-tion and absorption correction. The absorption coefficient obtained from thefluorescence spectrum is shown in the right panel.

400

300

200

100

0

Inte

nsity

(cn

ts/s

)

-0.1 0 0.1

Dq PR (deg.)

0 kG

x = 0.06 (L) T = 1.5 K

27 Ð q

L =

27 + q

(a) (0, 0, L )

-0.1 0 0.1

Dq PR (deg.)

9.9 kG // b*

E = 8.934 keV (E2)

27 Ð 2q

L =

27 + 2q

(b)

Fig. 12. (Color online) ∆θPR dependence of the E2 resonance intensity for(a) (0, 0, 27 ± q) at 0 kG and (b) (0, 0, 27 ± 2q) at 9.9 kG. The vertical dashedlines represent the positions of the LCP and RCP states. Solid lines are thefits using Eq. (B·5).

of the magnetic dipole, i(ε′ × ε) · Z(1)dip, as explained in §3.2.

The calculated Q-dependences of the parameters are shownby the lines in Fig. 13(a).

The problem is that the results of the E2 resonance inFigs. 12(a) and 13(b) cannot be reproduced by the scatteringfrom the magnetic dipole moment. Using Eq. (B·10), the E2scattering-amplitude matrix for the magnetic dipole with thestructure factor (1,±i, 0) is expressed as

FE2 =

(∓ sin 2θ i cos 3θ−i cos 3θ ∓2 sin 4θ

). (5)

For Q = (0, 0, 27 ± q), when φ = π/2 for the R-crystal inthe present case, C2/C0 = ∓0.79 and C3/C0 = 0.51 are ob-tained by calculation. However, the sign of C2 is opposite tothe experimental result. In addition, since the cos 3θ in Fπσ′and Fσπ′ changes sign at θ = π/6, C2 in Fig. 13(b) shouldchange sign around L=20, which clearly disagrees with theexperimental result.45) Therefore, we cannot explain the E2resonance by only the scattering from the magnetic dipole.

The Q-dependence of C2 and C3 for E2 can be reproducedby considering the scattering from the magnetic octupole. Weuse the following model. First, the structure factor of the mag-

3020100

L (r.l.u.)

8.934 keV (E2)

C2 (+q)

C2 (Ð q)

C3 (Ð q,+q)

(b)

-1.0

-0.5

0.0

0.5

1.0

C2/C

0, C

3/C

0

3020100

L (r.l.u.)

8.944 keV (E1)

x = 0.06 (R)0 kG

C2 (+q)

C2 (Ð q)

C3 (Ð q,+q)

(a)

(0, 0, L)L = 3n ± q

Fig. 13. (Color online) L (= 3n ± q) dependence of the parameters C2/C0and C3/C0 at zero field obtained from the ∆θPR scans for the x=0.06 (R) sam-ple at (a) E1 resonance and (b) E2 resonance. Solid, dashed, and dot-dashedcurves represent the calculations of C3/C0 (L = 3n± q), C2/C0 (L = 3n− q),and C2/C0 (L = 3n + q), respectively, for (a) E1 resonant scattering fromhelimagnetic dipole order and (b) E2 resonant scattering from helimagneticdipole and octupole order.

netic dipole (rank-1) for Q = (0, 0, 3n ± q) for the R-crystal isrepresented by

Z(1)dip = (1,±i, 0) . (6)

Second, we take into account the Tαx,y,z (T1u) and T βx,y,z (T2u)magnetic octupole (rank-3). The structure factors of these mo-ments, which are compatible with the helical magnetic dipoleorder, should be written as

Z(3)α =

√3Aeiϕ(1,±i, 0) , (7a)

Z(3)β =

√5Aeiϕ(1,∓i, 0) , (7b)

where A and ϕ represent the relative amplitude and phase, re-spectively, of the octupolar structure factor with respect tothose of Z(1)

dip. By putting A = 0.46 and ϕ = −0.79π, thedata are well explained as shown by the calculated curves inFig. 13(b). The ∆θPR dependence in Fig. 12(a) can also bereproduced by using these parameters.

The detailed relationship of the coefficients betweenEqs. (7a) and (7b) originates from the irreducible representa-tion of the order parameter in the D3 point group.46) Althoughwe use the cubic coordinates for the intensity calculation, weshould analyze the order parameter using the hexagonal co-ordinates. The in-plane dipole moments, Jx and Jy, for exam-ple, belong to the two-dimensional Eu representation in theD3 point group. The octupole moments with the Eu represen-tation, on the other hand, are expressed as linear combinationsof Tα and T β. They are written as T γx = (−

√3Tαx −

√5T βx ) and

T γy = (−√

3Tαy +√

5T βy ). Therefore, the ordered structure ofthe octupole moments given by Eqs. (7a) and (7b) is equiv-alent to the helical octupole ordering expressed by T γx + iT γy ,which has the same symmetry as the helical magnetic dipoleorder expressed by Jx+iJy. Although the opposite sign of Z(3)

β,yin Eq. (7b) is a rather tricky result, it is not surprising.

Other representations are given by T δx = (√

5Tαx −√

3T βx )(A2u) and T δy = (

√5Tαy +

√3T βy ) (A1u). Txyz and T βz also con-

stitute another Eu representation in the D3 group. Although

7

J. Phys. Soc. Jpn.

Fig. 14. (Color online) Magnetic charge distributions representing (a) amagnetic dipole Jx, (b) a magnetic octupole −T γx , (c) the superposition ofJx and −T γx . (d) Charge and magnetic charge density of the |7/2,±1/2⟩ stateof Yb3+, possessing a magnetic dipole moment ⟨Jx⟩.47)

these moments generally need to be included, the data weresuccessfully explained without taking them into account.

4. Discussion4.1 Helical order of octupole moments

The ground multiplet of J = 7/2 for Yb3+ splits into fourKramers doublets in the crystalline electric field of YbNi3Al9,represented by the point group D3. Since the magnetic mo-ments preferentially lie in the c plane, the crystal field groundstate is likely to be composed mainly of | 72 ,±

12 ⟩, which is

isotropic in the c plane. In this two-dimensional space of| 72 ,±

12 ⟩, the x-component matrices of Jx (Eu dipole) and T γx

[x(5z2−r2)-type, Eu octupole] are expressed in the same formas

Jx = 2(0 11 0

), T γx = −15

√32

(0 11 0

). (8)

Therefore, when a magnetic dipole moment ⟨Jx⟩ is induced,the ⟨T γx ⟩ octupole arises simultaneously. When a helical mag-netic dipole order of ⟨Jx+iJy⟩ occurs, the helical magnetic oc-tupole order of ⟨T γx + iT γy ⟩ with the same helicity also arises.The matrix elements of the Txyz and T βz octupoles (Eu), T δx(A2u), and T δy (A1u) all vanish in the | 72 ,±

12 ⟩ space. This is

the reason we did not need to include these moments in theanalysis in §3.6. Also, there is no degree of freedom for thequadrupole (rank-2) moments, which guarantees the reason-ing that the orbital contribution is not included in the second-harmonic signal.

The T γ-type magnetic octupole reflects an asphericalcharge distribution of the | 72 ,±

12 ⟩ states. Figure 14 shows

schematics of the magnetic charge distributions.47) The su-perposition of the Jx-dipole and the T γ-octupole, shown inFigs. 14(a) and 14(b), respectively, results in the magneticcharge distribution shown in Fig. 14(c). We see a lower mag-netic moment density along the c (z) axis, which is associatedwith the lower charge density of the | 72 ,±

12 ⟩ state along the c

axis as shown in Fig. 14(d).At the present stage, it is not clear whether the octupole

moments affect any physical property in Yb(Ni1−xCux)3Al9.It should be remarked, however, that the strength of the inter-ionic multipolar interaction is rank-independent in the RKKYmechanism.48, 49) As a result, the higher-rank multipoles playimportant roles in the physical properties and can equally bea primary order parameter, as is typically realized in the cu-bic CexLa1−xB6 system, in which the dipole, quadrupole, andoctupole moments are active as independent degrees of free-dom.50, 51) In Yb(Ni1−xCux)3Al9, by contrast, the active mo-ments in the doublet ground state are (Jx, Jy) or (T γx ,T

γy ),

which both belong to the Eu representation and are not in-dependent. The issue is how much the physical properties are

affected by the octupolar interaction in comparison with thedipolar interaction.

4.2 Temperature-dependent helical magnetic structureAs shown in Fig. 7, the propagation vector changes with

temperature by an amount proportional to the magnitude ofthe ordered moment. This suggests that the RKKY exchangeinteraction itself changes with the evolution of the orderedmoment.52) The magnetic propagation vector generally re-flects the q vector where the exchange interaction J(q), theFourier transform of Ji j, takes a maximum. J(q) for theRKKY interaction is associated with the local c- f exchangeinteraction and χ(q) for the conduction electron system. Un-fortunately, little is known yet about the Fermi surface ofYb(Ni1−xCux)3Al9. When a helimagnetic order develops onthe Yb sites, a perturbation of the exchange field to the con-duction electron system arises, which is also described by thesame q vector. As a result, a gap appears in the region ofthe Fermi surface where εk′ = εk+q is satisfied.53) This gapslightly modifies χ(q) and J(q), resulting in a shift of the qvector.54) The q-shift from the original value of q0 just belowTN becomes almost proportional to the ordered moment. Asimilar T dependence of the q vector has also been reportedfor GdSi, GdNi2B2C, and GdPd2Al3.55–58) In any case, con-cerning the most noteworthy phenomenon that the directionof the q-shift differs below and above x=0.03, we have no ex-planation yet. Knowledge of the band structure and the Fermisurface is required.

If there were magnetic anisotropy, the T dependence of theq vector would be more complicated as observed in rare-earthmetals.52) An anisotropy in the c plane would cause a squaringup and the appearance of the third-harmonic peak. This effectalso causes the q vector to shift from q0. However, in sucha case, the q-shift becomes proportional to (TN − T )2, whichis different from the present case of Yb(Ni1−xCux)3Al9.52, 59)

Furthermore, the |7/2,±1/2⟩ states in Yb(Ni1−xCux)3Al9 arealmost isotropic in the c plane.

4.3 Formation of chiral soliton latticeThe results at zero field (Figs. 3 and 4) show that the helic-

ity of the helimagnetic structure in Yb(Ni1−xCux)3Al9 has aone-to-one relation with the crystal chirality. This shows thata DM exchange interaction indeed exists, lifting the degener-acy between the left- and right-handed helical magnetic struc-tures. The situation is similar to the well-investigated com-pound CrNb3S6, and Yb(Ni1−xCux)3Al9 can also be recog-nized as a monoaxial chiral helimagnet. The shift of the q-vector and the appearance of the higher-harmonic peaks inmagnetic fields (Figs. 8–10) suggest the formation of a CSL,a characteristic outcome of a monoaxial chiral helimagnet.

Theoretical study on the CSL state has been performed us-ing the chiral sine-Gordon model, which has been success-fully applied to CrNb3S6.10) For CrNb3S6, the application ofthe sine-Gordon model, where continuous variables are intro-duced to treat the spin arrangement, is justified because themodulation length of the helimagnetic structure L0 (∼ 480 Å)is much larger than the interlayer distance c0 =12.1 Å. On theother hand, in Yb(Ni1−xCux)3Al9, it should be remarked thatthe longest L0 of 61 Å for x=0.06 is only 6.7 times larger thanc0=9.1 Å (=c/3), which questions the simple application ofthe continuous model. However, without an appropriate the-

8

J. Phys. Soc. Jpn.

1.0

0.9

0.8

0.7

q (H

) / q

(0)

1.00.50

H / HC0

x = 0.06

(a)

SG model

0.3

0.2

0.1

0

Inte

nsity R

atio

1.00.50

H / HC0

I2q / Iq

I3q / Iq

(b)

Fig. 15. (Color online) (a) Comparison of the experimental data ofq(H)/q(0) for x = 0.06 with the L0/LCSL curve obtained from the chiralsine-Gordon (SG) model.10, 11) µ0HC0 is set to 11.5 kG. (b) Comparisonof the experimentally obtained intensity ratios of I2q/Iq (filled circles) andI3q/Iq (open square) with the calculated ratio for the sine-Gordon model us-ing µ0HC0=10.5 kG.

c

a

b

H / HC0 = 0(a)L = 6.7c0

H / HC0 = 0.4(b)L = 7c0

a

b

c

H / HC0 = 0.8(c)L = 8c0

a

b

c

L

H(d)

Fig. 16. (Color online) (a)–(c) Magnetic structures of Yb(Ni1−xCux)3Al9for x=0.06 (L) in magnetic fields expected from the chiral sine-Gordonmodel. Only the moments on Yb-1 sites are shown. (a) Helimagnetic orderedstate at zero field with q=0.445 and the helical pitch of L=6.7c0. This isthe same as Fig. 5. (b) CSL state for H/HC0=0.4, corresponding to q=3/7and L=7c0 at around 5 kG. (c) CSL state for H/HC0=0.8, corresponding toq=3/8 and L=8c0 between 9.0 and 9.5 kG. (d) CSL state for continuous one-dimensional spin model corresponding to (c), H/HC0=0.8.

ory at the present stage, it is worth comparing the results withthe sine-Gordon model.

In Fig. 15(a), we compare the H dependence of the nor-malized propagation vector q(H)/q(0) for x = 0.06 with thecalculation of L0/LCSL for the sine-Gordon model.10, 11) Weintroduced a critical field of µ0HC0=11.5 kG so that the datafit the calculation. The data points closely follow the theoret-ical curve up to H/HC0 ∼ 0.85, where the first-order tran-sition to the ferromagnetic state takes place. In Fig. 15(b),we compare the intensity ratios of I2q/Iq and I3q/Iq with thetheoretical curve by using a slightly different µ0HC0 of 10.5kG. These intensity ratios also closely follow the theoreti-cal curve. Although the calculation is for neutron diffraction,

the result can almost equally be applied to the intensity ofRXD. These agreements between the experiment and theorystrongly indicate that the CSL state is actually formed in theYb(Ni1−xCux)3Al9 system.

It is a problem, however, that the first-order transition atH/HC0 ∼ 0.85 is not explained by the theory, probably be-cause the sine-Gordon model is too simple to describe the ac-tual magnetic exchange interactions in the Yb(Ni1−xCux)3Al9system. It may be associated with the short period of theoriginal helimagnetic structure. It is interesting to comparethe Yb(Ni1−xCux)3Al9 system with the long-period system ofCrNb3S6, in which the CSL state survives against the ferro-magnetic state up to H/HC0 ∼ 0.97.12) Also, the lock-in phe-nomenon is not explained because the coupling with the lat-tice, other than the basal plane anisotropy, is not included inthe theory. At the same time, we need to keep in mind whythe continuous model of the sine-Gordon theory appears to beso applicable to the short-period system of Yb(Ni1−xCux)3Al9below H/HC0 ∼ 0.85. This is an important problem to bestudied in future.

Putting aside the applicability of the theory, we show inFigs. 16(a)–16(c) the theoretically expected magnetic struc-tures in magnetic fields for x=0.06 (L). Of special inter-est is the locked-in structure with q=3/8 between 9 and 9.5kG, which is shown in Fig. 16(c). Another structure withq=3/7=0.4286, which is expected to appear at around 5 kG,is shown in Fig. 16(b) for comparison. Since it may be dif-ficult to understand the CSL structure from these figures be-cause of the discontinuity in the spin arrangement, we showfor reference in Fig. 16(d) the CSL state for a continuous one-dimensional spin model corresponding to the CSL state inFig. 16(c). The CSL structure in Fig. 16(c) gives rise to allthe harmonics of q, 2q, 3q, · · · . It also reproduces the ∆θPR-scan data for the 2q peak shown in Fig. 12(b) because thestructure factor is written in the same way as (1,±i, 0). Theseconsistencies with the data suggest that the CSL structure inFig. 16(c) is actually realized. If this is the case, there shouldbe no reason why the q=3/7 structure of Fig. 16(b) is not stabi-lized. Experimentally, however, the q=3/7 structure does notseem to be locked in at around 5 kG as observed in Fig. 9.This shows that the lock-in behavior is strongly associatedwith the CSL formation. The CSL with q=3/8 at 9 kG locksin the lattice because the CSL is well developed, whereas theCSL with q=3/7 is not locked in because it is still weak at 5kG.

4.4 Magnetic exchange and DM interactionsThere are many problems concerning the magnetic ex-

change and DM interactions in this system. The sine-Gordontheory is based on the following one-dimensional model fora layered monoaxial chiral helimagnet with ferromagneticnearest-neighbor exchange and DM interaction:

H = −J1

∑i

Si · Si+1 − D ·∑

i

Si × Si+1 −∑

i

Si · H , (9)

where Si lies in the xy plane, D is along the z axis, and H isapplied in the xy plane. At H = 0, a helimagnetic order withq0 = c−1

0 arctan(D/J1) is formed, where c0 is the interlayerdistance. If we simply apply this model to Yb(Ni1−xCux)3Al9to obtain q0 values of 0.42, 0.54, and 0.63 for x=0.06, 0.04,and 0.02, D/J1 is required to be 1.2, 2.1, and 3.9, respectively,

9

J. Phys. Soc. Jpn.

which are unrealistically large. For x=0, in addition, it is im-possible to reproduce q0=0.82 from this model because theangle between spins on neighboring layers is larger than 90.To explain these large q0 values, it is necessary to includemore long-range exchange interactions mediated by the con-duction electrons. It is more reasonable to consider that q0 isdetermined mainly by J1 (> 0), J2 (< 0), and further interac-tions, with which J(q) takes a maximum at q0. If we considerJ1 and J2, for example, J(q) = J1 cos qc0 + J2 cos 2qc0, andq0 is given by c−1

0 arccos(−J1/4J2). The degeneracy of ±q0 islifted by the DM term.

However, there still remains a problem. If q0 is determinedby the maximum of J(q), it is expected that TN ∝ J(q0) andHC ∝ J(q0) − J(0). When q0 approaches 0 (ferromagneticstate) with increasing Cu concentration, J(q0) − J(0) shouldalso become small because J(q) is a symmetric function ofq.60) However, while TN=6.5 K for x=0.06 is twice as largeas that for x =0, µ0HC=10 kG for x=0.06 is ten times aslarge as that for x=0. These facts are not explained by thesimplistic model above and show that the situation behindthe helimagnetic order of Yb(Ni1−xCux)3Al9 is not so sim-ple. Firstly, the interaction between Yb-1 and Yb-2 momentsshould be considered. More importantly, the fact that HC sig-nificantly increases with increasing x, i.e., with decreasing q0and J(q0)− J(0), suggests that the nonlinear effect of the CSLformation in magnetic fields plays an important role in keep-ing the helimagnetic structure and preventing the system fromundergoing a transition to the ferromagnetic state. The pos-sibility of the DM interaction increasing with increasing Cuconcentration should also be considered.

5. SummaryWe performed resonant X-ray diffraction experiments to in-

vestigate the helical magnetic structure of Yb(Ni1−xCux)3Al9with the space group R32, which lacks both inversion andmirror symmetries and allows the existence of left- and right-handed crystal structures.• We showed that the magnetic structure is described by

an incommensurate propagation vector (0, 0, q), whereq=0.818, 0.617, 0.553, and 0.445 for x=0, 0.02, 0.04,and 0.06, respectively, at the lowest temperature. Themagnetic moments lie in the c plane.• We clarified that the helicity of the magnetic structure

has a one-to-one relation with the crystal chirality, indi-cating that an antisymmetric exchange interaction medi-ated by the conduction electrons exists, i.e., the RKKYmechanism. A theoretical study is required to providefurther insight into the physical mechanism.• When a magnetic field is applied perpendicular to the

helical axis, the q value decreases (the helical pitch in-creases) and, simultaneously, the higher-harmonic peaksof 2q and 3q are induced with increasing field. This be-havior, especially for the x=0.06 case, coincides withthe calculation for the chiral sine-Gordon model, whichdescribes the formation of the chiral soliton lattice(CSL) state in a monoaxial helimagnet. This coinci-dence provides strong evidence for the CSL formationin Yb(Ni1−xCux)3Al9, which has been suggested by mag-netization measurement. However, since the helical pitchis not much longer than the interlayer spacing, it is ques-tionable if the spin arrangement could be approximated

as a continuous medium. The applicability of the sine-Gordon model to Yb(Ni1−xCux)3Al9 remains a question.The lock-in phenomenon at q = 3/8 and the first-ordertransition to the ferromagnetic state also are importantissues to be studied.

• We observed an anomalous temperature dependence ofthe propagation vector. The q value changes with tem-perature by an amount almost in proportion to the mag-netic order parameter, which we ascribed to the changein the exchange interaction with the development of theordered moment. It is not understood, however, why theq value decreases with decreasing T for x < 0.03, butthis is reversed for x > 0.03.

• The helical ordering of the magnetic octupole momentwas detected, which arises simultaneously with the he-lical ordering of the magnetic dipole moment, reflectingthe anisotropic charge density of the crystal field groundstate of Yb3+, consisting mainly of |7/2,±1/2⟩.

AcknowledgementsWe acknowledge valuable comments by H. Ohsumi, M.

Suzuki, T. Nagao, R. Shiina, A. Leonov, and J. Kishine. Thiswork was supported by JSPS KAKENHI Grant Numbers15K05175, 26400333, 16H01073, 15H05885, 17H02912,and 16KK0102. The synchrotron radiation experiment wasperformed under Proposals No. 2015B3711, No. 2016A3761,No. 2016B3762, and No. 2017A3787 at BL22XU of SPring-8. We also acknowledge support by the JSPS Core-to-CoreProgram, A. Advanced Research Networks. Magnetizationmeasurement with the MPMS and the electron probe micro-analysis were performed at N-BARD, Hiroshima University.

Appendix A: Confirmation of the Crystal StructureA.1 Crystal chirality

The crystal chirality of the sample, especially the spotwhere the X-ray beam is irradiated in the RXD experiment,can be confirmed at the beamline by measuring the energydependence of an appropriate fundamental Bragg peak asshown in Fig. A·1. The structure factor of the (1, 1, 24) fun-damental reflection for the right-handed crystal is expressedas FR,(1,1,24)(ω) = AAl fAl(ω) + ANi fNi(ω) + AYb fYb(ω), whereAAl = 0.24 + 24.6i, ANi = −8.6 + 14.9i, and AYb = 6.0.FL,(1,1,24) also has the same expression. For FL,(1,1,24) andFR,(1,1,24), we take the complex conjugates for AAl and ANi.When fYb(ω) = f0,Yb + f ′Yb(ω) + i f ′′Yb(ω) exhibits an anoma-lous dispersion around the absorption edge, the intensity ofthe Bragg reflection also exhibits a strong energy dependence.Since the intensities for the right- and left-handed crystals areexpressed as |FR,(1,1,24)(ω)|2 and |FL,(1,1,24)(ω)|2, respectively,they exhibit different energy dependences due to the differentinterference effect. As shown in Fig. A·1, the relative relationof the intensity is reversed for the (1, 1, 24) reflection. Therelationship of the intensity is consistent with the calculatedspectrum assuming the predetermined crystal chirality.

A.2 Threefold symmetry about the c axisFigure A·2 shows the rocking scans of the (1, 0, L) reflec-

tions for the x=0.06 (L) sample. Only the (1, 0, 3n + 1) re-flection is allowed in the (1, 0, L) reflections in the R32 spacegroup. These data guarantee the threefold symmetry of this

10

J. Phys. Soc. Jpn.

sample about the c axis. If the forbidden reflection is ob-served, it means that the [1 1 0] axis is mixed with the a axisdue to the stacking fault of 60.

Appendix B: Formalism of Resonant X-ray ScatteringWe use the scattering-amplitude-operator method to ana-

lyze the experimental results.42) We consider a 2×2 matrix F,consisting of four elements of the scattering amplitudes forσ-σ′, π-σ′, σ-π′, and π-π′:

F =(Fσσ′ Fπσ′Fσπ′ Fππ′

). (B·1)

By using the identity matrix I and the Pauli matrix σ, F cangenerally be expressed as

F = βI + α · σ =(β + α3 α1 − iα2α1 + iα2 β − α3

). (B·2)

Once we know the matrix F, the scattering cross section(dσ/dΩ) can be calculated by(

dσdΩ

)= TrµF∗F

= β∗β + α∗ · α + β∗(P · α) + (P · α∗)β+ iP · (α∗ × α) , (B·3)

150

100

50

0

Inte

nsity

(arb

. un

it)

9.008.958.908.85

X-ray Energy (keV)

x = 0.06 (R)

9.008.958.908.85

x = 0.06 (L)

(Ð1, Ð1, 24) (1, 1, 24)

Fig. A·1. (Color online) X-ray energy dependence of the (1, 1, 24) and(1, 1, 24) fundamental reflections for the right- and left-handed crystals forx=0.06 around the L3 absorption edge of Yb. Solid lines are the guides forthe eye.

60

50

40

30

20

10

0

Inte

nsity (a

rb. unit)

37.537.036.536.0

w (deg.)

Yb(Ni1-xCux)3Al9x = 0.06 (L)(1 0 16)

x1/500

(1 0 17) (1 0 18)

Fig. A·2. (Color online) Rocking scans of the (1, 0, 16), (1, 0, 17), and(1, 0, 18) reflections for the x=0.06 (L) sample.

where µ = (I + P · σ)/2 represents the density matrix andthe Stokes vector P = (P1, P2, P3) represents the polarizationstate of the incident photon. P1, P2, and P3 represent the de-grees of +45 (P1 = 1) or −45 (P1 = −1) linear polarization,right (P2 = 1) or left (P2 = −1)-handed circular polarization,and the σ (P3 = 1) or π (P3 = −1) linear polarization state,respectively.

Using the elements of Eq. (B·1), the scattering cross sectionis expressed as(

dσdΩ

)=

12( |Fσσ′ |2 + |Fσπ′ |2 + |Fπσ′ |2 + |Fππ′ |2 )+ P1Re

F∗πσ′Fσσ′ + F∗ππ′Fσπ′

+ P2Im

F∗πσ′Fσσ′ + F∗ππ′Fσπ′

(B·4)

+12

P3( |Fσσ′ |2 + |Fσπ′ |2 − |Fπσ′ |2 − |Fππ′ |2 )

.

Therefore, the cross section for the incident beam describedby (P1, P2, P3) can generally be written as(

dσdΩ

)= C0 +C1P1 +C2P2 +C3P3 , (B·5)

which can be used as a fitting function for the ∆θPR scan withthe four parameters of Cn (n = 0 ∼ 3).

The four elements of Eq. (B·1) at an X-ray energy E = ℏωnear the resonance energy are expressed as

Fεε′ = FE1,εε′ (ω) + FE2,εε′ (ω) . (B·6)

FE1,εε′ (ω) and FE2,εε′ (ω) for a scattering process from thephoton state (ε, k) to (ε′, k′) are expressed as

FE1,εε′ (ω) =2∑ν=0

α(ν)E1(ω)

∑Γ

X(ν)E1,Γ(ε, ε

′) · ⟨Z(ν)Γ⟩ , (B·7)

FE2,εε′ (ω) =4∑ν=0

α(ν)E2(ω)

∑Γ

X(ν)E2,Γ(ε, ε

′, k, k′) · ⟨Z(ν)

Γ⟩ .

(B·8)

⟨Z(ν)Γ⟩ represents the structure factor of the rank-ν multipole

moment of the irreducible representation Γ in the cubic coor-dinates. X(ν)

Γis the geometrical factor corresponding to ⟨Z(ν)

Γ⟩.

For the rank-1 dipole moment with the T1u representation,

X(1)E1,dip =

i√

2(ε′ × ε) , (B·9)

X(1)E2,dip =

i

2√

10

(k′ · k)(ε′ × ε) + (ε′ · ε)(k′ × k)

+ (k′ · ε)(ε′ × k) + (ε′ · k)(k′ × ε) . (B·10)

See Ref. 36 for X(3)E2. The scattering amplitude is obtained by

taking the scalar product of X(ν)Γ

and ⟨Z(ν)Γ⟩.

The same result is obtained by using the spherical tensormethod.35) By transforming ⟨Z(ν)

Γ⟩ in the cubic representation

to ⟨T (ν)q ⟩ in the spherical representation,

FE1,εε′ (ω) =2∑ν=0

α(ν)E1(ω)

ν∑q=−ν

(−1)qX(ν)E1,−q⟨T

(ν)q ⟩ , (B·11)

11

J. Phys. Soc. Jpn.

FE2,εε′ (ω) =4∑ν=0

α(ν)E2(ω)

ν∑q=−ν

(−1)qX(ν)E2,−q⟨T

(ν)q ⟩ , (B·12)

where the geometrical factors of X(ν)E1,q and X(ν)

E2,q are also ex-pressed in the spherical representation as

X(ν)E1,q =

1∑p,p′=−1

ε′pεp′⟨1p1p′|νq⟩ , (B·13)

X(ν)E2,q =

2∑p,p′=−2

h′p′hp⟨2p2p′|νq⟩ , (B·14)

hm =

1∑p,p′=−1

εpkp′⟨1p1p′|2m⟩ , (B·15)

h′m =1∑

p,p′=−1

ε′pk′p′⟨1p1p′|2m⟩ . (B·16)

It is important that the spectral functions of α(ν)E1(ω) and

α(ν)E2(ω) are rank-dependent.36, 37) In the data analysis, each of

them may be approximated by the form

α(ν)(ω) =eiϕν

ℏω − ∆ + iγ, (B·17)

where ∆, γ, and ϕν are the resonance energy, lifetime broad-ening effect, and phase factor, respectively.

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60) (Supplemental material) A simple calculation assuming J1 and J2 isprovided online in Fig. S2.

13

J. Phys. Soc. Jpn.

-1.0

-0.5

0.0

0.5

1.0

C2/C

0, C

3/C

0

3020100

L (r.l.u.)

rank-1 only (a)

3020100

L (r.l.u.)

(b)

C2 (Ð q)

C3 (Ð q,+q)

rank-3 only

C2 (+q)

8.934 keV (E2), x = 0.06 (R), (0, 0, 3n ± q), 0 kG

Fig. S·1. Comparison of the disagreeing calculation with the L (= 3n ± q) dependence data of the parameters C2/C0 and C3/C0 at zero field, which wasobtained from the ∆θPR scans for the x=0.06 (R) sample at the E2 resonance energy. Solid, dashed, and dot-dashed curves represent the calculation of C3/C0(L = 3n ± q), C2/C0 (L = 3n − q), and C2/C0 (L = 3n + q), respectively, for models of (a) ⟨Jx ± iJy⟩ dipole (rank-1) only and (b) ⟨T γx ± iT γy ⟩ octupole (rank-3)only. The data can be explained by mixing these two scattering contributions as described in the main text.

-3

-2

-1

0

1

J (q

)

1.51.00.50.0

q (r.l.u.)

x = 0 x = 0.02 x = 0.04 x = 0.06

2

1

0

-1

J1, J

2

6420

x (%)

J1

J2

Fig. S·2. A simple calculation of J(q) for one dimensional spin chain with nearest-neighbor exchange J1 and next-nearest-neighbor exchange J2. J1 andJ2 are taken so that q0 = c−1

0 arccos(−J1/4J2), where J(q) = J1 cos qc0 + J2 cos 2qc0 takes the maximum, reproduces the experimental value and J(q0) isproportional to TN. The degeneracy at ±q0 will be lifted by including the DM term. In this model, without considering any nonlinear effect such as CSLformation, HC will be proportional to J(q0) − J(0). However, it is contradictory to the experimental result.

Table S·1. Irreducible representation (Irrep.), operator notation, and basis functions of the dipole (rank-1) and octupole (rank-3) moments in the Oh and D3point groups.

Oh D3

Rank Irrep. Notation Basis Function Irrep. Notation Basis Function

1 T1u Jx x Eu Jx x

Jy y Jy y

Jz z A2u Jz z

3 T1u Tαx x(5x2 − 3r2)/2 Eu T γx = (−√

3Tαx −√

5T βx )/2√

2√

6x(5z2 − r2)/4

Tαy y(5y2 − 3r2)/2 T γy = (−√

3Tαy +√

5T βy )/2√

2√

6y(5z2 − r2)/4

Tαz z(5z2 − 3r2)/2 A2u Tαz z(5z2 − 3r2)/2

T2u T βx√

15x(y2 − z2)/2 A2u T δx = (√

5Tαx −√

3T βx )/2√

2√

10x(x2 − 3y2)/4

T βy√

15y(z2 − x2)/2 A1u T δy = (√

5Tαy +√

3T βy )/2√

2√

10y(y2 − 3x2)/4

T βz√

15z(x2 − y2)/2 Eu T βz√

15z(x2 − y2)/2

A2u Txyz√

15xyz Txyz√

15xyz

14