Correlation between spin helicity and electric polarization vector in quantum-spin chain magnet LiCu$_2$O$_2$

0.52 MB·English

by  S. Seki

Correlation between spin helicity and electric polarization vector in quantum-spin
chain magnet LiCu O
2 2
S. Seki1, Y. Yamasaki1, M. Soda2, M. Matsuura2, K. Hirota2 and Y. Tokura1,3
1 Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
2 The Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan
3 Multiferroics Project, ERATO, Japan Science and Technology Agency (JST), Tokyo 113-8656, Japan
(Dated:)
Measurements of polarized neutron scattering were performed on a S = 1/2 chain multiferroic
LiCu2O2. In the ferroelectric ground state with the spontaneous polarization along the c-axis, the
8 existenceoftransversespiral spin component in thebc-planewas confirmed. Whenthedirection of
0 electricpolarization isreversed,thevectorspinchiralityasdefinedbyCij =Si×Sj (iandj being
0 theneighboringspinsites) isobservedtobereversed,indicatingthatthespin-currentmodelorthe
2 inverseDzyaloshinskii-Moriyamechanismisapplicableeventothiseg-electronquantum-spinsystem.
n Differentialscatteringintensityofpolarized neutronsshows alarge discrepancy from thatexpected
a for theclassical-spin bc-cycloidal structure,implying the effect of large quantumfluctuation.
J
PACSnumbers: 75.80.+q,75.25.+z,75.10.Pq,77.22.Ej
6
1

Magnetoelectric effect, controlling dielectric (mag- have been proposed; the model based on the symmetric
]
l netic) properties by magnetic (electric) field, has been exchange interaction (S S ) or the antisymmetric one
e i j
·
- studiedfor longbecauseofits potentialfornovelphysics (S S )[4]. Forthelattercase,amicroscopicmodelwas
i j
r ×
t and application. Although several low-symmetric mate- devised by Katsura, Nagaosa and Balatsky (KNB)[5],
s
rials like Cr O were found to show the linear magneto- in which the electric polarization P produced between
. 2 3 ij
t
a electric response, the effect has been very small[1]. Re- magnetic moments at neighboring sites i and j (mi and
m cently, the phenomenon of electric polarizationflop with m ) is given as
j
- magneticfieldwasfoundforperovskitetypeTbMnO3[2].
d In this material,the specific magnetic structure itself in- Pij =A eij (mi mj) (1)
n · × ×
duces ferroelectricity, which enables the colossal magne-
o
Here, e is the unit vector connecting the site i and j,
c toelectric responsesvia the magnetic phasetransition[3]. ij
and A a coupling constant related to the spin-orbit and
[ Thekeyissueisthecouplingmechanismbetweenthespin
habit andthe polarization. Thus far,mainly two models exchange interactions. This model predicts that a heli-
1
magnet with transverse spiral components can be ferro-
v
electric, and well explains the ferroelectric behaviors ob-
3
3 servedforRMnO3(R=TbandDy)[2,6,7],Ni3V2O8[8],
5 CoCr O [9], MnWO [10], etc. All these materials con-
2 4 4
2 tain the frustration of magnetic interactions as a source
.
1 of noncollinear spin structure. Besides, the ferroelectric
0 spiralmagnets based onother than KNB model, such as
8 CuFeO [11, 12, 13, 14], have also been reported. These
2
0
materials with both magnetic and dielectric orders are
:
v now broadly termed multiferroics.
i
X LiCu2O2,asinvestigatedhere,hasrecentlybeenfound
to be one such member of multiferroics[15]. Figure 1(a)
r
a indicates the crystal structure of LiCu O ; the space
2 2
group Pnma, and lattice parameters a = 5.73, b = 2.86
and c = 12.4˚A at room temperature[16]. This material
contains equal number of Cu1+ and Cu2+, only the lat-
terofwhichcarriesspinS=1/2. EachCu2+ ionisonthe
center of oxygen square and forms edge-shared chains
running along the b-axis with the Cu-O-Cu bond angle
◦
of 94 . As expected from the Kanamori-Goodenough
FIG. 1: (color online). (a)Crystal structure of LiCu2O2. (b) rule, the nearest neighbor exchange interaction (J ) is
Schematic view of magnetic interactions between Cu2+ sites. 1
ferromagnetic though relatively weak as compared with
(c),(d) Temperature dependence of magnetic susceptibility,
electricpolarizationanddielectricconstant. Allthequantities the antiferromagnetic next nearest neighbor interaction
were measured in thewarming process. (J2), causing the magnetic frustration. The magni-
tude of inter-chain interaction (J ) is presumed to be
DC
2
small(< J , J ),thoughhasnotreachedtheconsensus for the neutron study is 12 mm2 (ab plane) 0.6 mm (c-
1 2
| | | | ×
as yet (Fig. 1 (b))[17, 18]. As a result of the frustration, axis). Allthedatapresentedinthispaperweremeasured
a spiral magnetic structure is realized below T 23K. on the identical sample. Dielectric constant was mea-
N2
∼
A former (unpolarized) neutron diffractionstudy has re- sured at 100kHz using an LCR meter. For the electric
vealed the incommensurate magnetic structure with the polarization, we measured the pyroelectric current with
modulation vector (0.5, 0.174, 0), and claimed the ab- a constant rate of temperature sweep ( 2K/min) and
∼
spiral state[16]. In this phase, however, the appearance integrated it with time. To obtain a single ferroelectric
of spontaneous electric polarization along the c-axis has domain, the poling electric field was applied in the cool-
recentlybeenreported[15]. Toreconciletheobservedpo- ing process and removed just before the measurements
larization direction with the spiral spin state, the KNB of pyroelectric current and polarized neutron scattering.
model requires the bc-spiral spin structure. Recent reso- Magnetization was measured with a Magnetic Property
nantsoftx-raymagneticscatteringstudysuggestsamore Measurement System (Quantum Design Inc.).
complexspinspiral[19],andthemagneticstructureofthe Figures 1 (c) and (d) show the temperature depen-
ferroelectric groundstate is still under controversy. Inci- dence of magnetic susceptibility, dielectric constant, and
dentally, the powder neutron study on the isostructural electric polarization for LiCu O . For H c, the tem-
2 2
k
material NaCu2O2 justifies the bc-spiral spin structure, perature derivative of magnetic susceptibility (dχ/dT)
whilethemagneticmomentofCu2+ isestimatedassmall indicates two anomalies at T 24.5K and T 23.0K,
N1 N2
∼ ∼
as 0.56µB[20]. This implies that the effect of quantum although only one peak at TN2 is found in dχ/dT for
fluctuation is important also in LiCu2O2. H b (or a). These imply the existence of two mag-
k
In this paper, to clarify the originof ferroelectricityin netic phases below T ; AF1 (T > T >T ) and AF2
N1 N1 N2
LiCu O , we testify the validity of the KNB model for (T > T). The anomaly at 9K possibly caused by im-
2 2 N2
the e -electron spin system with potentially large quan- purity Li CuO [16, 22] was absent in our sample. The
g 2 2
tumfluctuation. Recently,thepolarizedneutronscatter- spontaneous electric polarization parallel to the c-axis
ing experiment on TbMnO has confirmed the coupling (P ) evolves only below T . The P can be reversed
3 c N2 c
between the spin vector chirality and the direction of withtheoppositepolingelectricfield(E ). Thisindicates
c
electric polarization in accord with the KNB model[21]. the ferroelectric nature of AF2 phase, and suggests the
Since the polarity-dependent vector chirality can be the correlation between ferroelectricity and magnetic prop-
definitiveevidenceforthespiral-spindrivenferroelectric- erties. Allthese featuresreproducedthe resultsreported
ity, we performed the related experiments on LiCu O . byS.Parketal[15,19],whoproposedthesinusoidalspin
2 2
Single crystals of LiCu O were grown by the self- structure with collinear spins (parallel to the c-axis) for
2 2
flux method. Under a polarized optical microscope, the AF1. A recent theory proposed the intriguing scenario
fine twin structure with mixing of the a and b-axis do- of the novel cholesteric spin state for this phase[23]. We
mains was observed in accord with the former observa- measured the poling electric field dependence of sponta-
tions [15, 22]. The crystal was cleaved into a thin plate neous polarization and confirmed that the saturation of
with the widest faces parallel to (001) plane. As the Pc was achieved above Ec 350kV/m. We also mea-
| | ∼
electrodes, Al was deposited on the ab faces. Polarized sured dielectric constant parallel to the c-axis (ǫc) and
neutrondiffractionexperimentswerecarriedoutwiththe found peaks at both TN1 and TN2, although previously
ISSP-PONTAtriple-axis spectrometer at JRR-3M using only one peak at TN2 was reported[15].
a Heusler polarizer. In this paper, we define the scat- For the polarized neutron diffraction measurements,
tering vector Q as Q = k - k, where k and k are we focused on the ferroelectric AF2 phase. Since differ-
f i i f
the wave vectorsof the incident and diffracted neutrons, entmagneticstructures,suchastheab-spiral[16]andthe
respectively. The polarization direction of incident neu- bc-spiral plus a-component structure[15, 19], have been
trons (S ), as defined by the magnetic field ( 10 mT) proposedforthisphase,whetherthemagneticmomentis
n
∼
generated with a Helmholtz coil, can be reversed by a presentalongthec-axiswasfirstexamined. Forthispur-
neutron-spin flipper. The polarized neutron scattering pose,wetooktheS Qsetup(Fig. 2(a)),whereneutron
n
⊥
experiments were executed for the two configurations; spinswereparallelorantiparalleltothec-axis. Todistin-
S Q and S Q (Figs. 2 (a) and (b)). The flipping guish between the spin-flip and non-spin-flip scattering,
n n
⊥ k
ratio of polarized to unpolarized neutrons measured at a Heusler analyzer was employed. In general, only the
the (2,1,0)nuclearreflectionwassufficientlylarge;33for magnetic moment perpendicular to Q contributes to the
S Q and 27 for S Q. The sample was mounted on magnetic reflection of neutrons. For polarized neutrons,
n n
⊥ k
a sapphire plate in a closed-cycle helium refrigerator, so furthermore, the magnetic moment parallel to S pro-
n
that the horizontal scattering plane of the spectrometer duces the non-spin-flip scattering and the moment per-
coincidedwiththe(hk 0)zone. Theneutronenergywas pendicularto S does the spin-flipscattering[24]. Figure
n
fixedat13.47meVandthecollimations40′ 40′ 40′ 80′ 2 (d) shows the k-scan profile of the (1.5, +δ, 0) mag-
− − −
were employed. Higher-order neutrons were removed by neticreflectionat7K(<T ). Theobservedmodulation
N2
a pyrolyticgraphitefilter. The size ofthe specimenused wavenumber, δ 0.175, is in accord with literature[16].
∼
3
FIG. 2: (color online). The experimental geometries for the
polarized neutron diffraction; (a) Sn⊥Q and (b) SnkQ. The
labels “on” and “off” indicate the state of the neutron-spin
flipper. (c) Schematic illustration of nuclear and magnetic FIG.3: (coloronline). (a)-(d)Thek-scanprofilesofthe(1.5,
Bragg positions in the reciprocal space. (d) The k-scan pro- ±δ, 0) magnetic reflections in the SnkQ setup. The labels
filesofthe(1.5,+δ,0)magneticreflectionintheSn⊥Qsetup. “on” and “off” show the state of neutron-spin flipper. Solid
lines show the result of the Gaussian fitting. (e), (f) The
geometricalrelationshipsbetweenspinchirality(helicity)and
electric polarization determined from theobserved results.
SinceQcanbeconsideredalmostparalleltothea-axisin
this configuration (Fig. 2(c)), the b-component of mag-
netic moment (mb) contributes to the spin-flip scatter- Here, ηi denotes the component of mi perpendicular to
ingwhilethec-component(mc)tothenon-spin-flipscat- Q, ηi = Qˆ (mi Qˆ), where Qˆ = Q/Q and Sˆn =
tering. Assuming the common background for the both S /S . Fo×r simpl×icity, we take hereafte|r t|he approxi-
n n
profiles,the integratedintensities arenearly equal(spin- mat|ion|that S Q a and define abc zxy, where z is
n
flip(mb)/non-spin-flip(mc) 0.9). Thissuggeststheex- thespinquantizkationkaxis. Then,thesp→invectorchirality
≈
istenceofthe nearlysameweightofb-andc-components onthe bc-planecanbe defined asC=(η η )/η η .
i j i j
inthemagneticstructureofAF2. Thisisconsistentwith With use of the relations η = (σx,σy,×0) and| σ×± =|
i i i
the bc-spiral (or plus some a-component) model[15, 19], σx iσy, the cross section for the (1.5, δ,0) magnetic
and at least not with the simple ab-spiralone[16]. refl±ections can be expressed as ±
Next, we attempted to observe the relationship be-
dσ dσ dσ
tween the polarizationdirection andthe chiralityof spin = (3)
spiral. For this purpose, we adopted the S Q setup (cid:16)dΩ(cid:17)± (cid:16)dΩ(cid:17)c±(cid:16)dΩ(cid:17)s
n
k
(Fig. 2(b)), where neutron spins are parallel or antipar-
where
allel to Q. In this alignment, only spin-flip scatterings
claorniztraitbiounteatnoaltyhseismisagnneeetdiecdr,eflanecdtiwone.eTmhpeloreyfeodret,hneotwpoo-- (cid:16)ddΩσ(cid:17)c ∝Xcos{Q(Ri−Rj)}·hσi+σj−i (4)
i,j
axes mode without an analyzer. Figures 3 (a)-(d) show
the k-scanprofilesof the (1.5, δ, 0) magnetic reflection
ac-ta7xKis[2(E5]cw).itEhcvwaraisouaspppolileidngaetl±3ec0tKric(>fieTlNd1s)paanradllreelmtoovthede (cid:16)ddΩσ(cid:17)s ∝Xi,j sin{Q(Ri−Rj)}·hσi+σj−i (5)
at7Kjust before the diffractionmeasurementsto obtain
asingleferroelectricdomain. With E =450kV/m,the For intuitive understanding, we tentatively treat the
c
| |
difference of intensity between δ was clearly observed, cross section in the classical limit. Based on the results
±
and the relative intensity was confirmed to be reversed for the S Q setup, we can assume the bc-spiral mag-
n
⊥
bychangingthe signofeither S orE . Thesebehaviors netic structure plus severala-component:
n c
can be interpreted in terms of the E -dependent vector
c
m =m e cos(q R )+m e sin(q R )
chirality of the transversebc-spiral spins as follows. i b b m i c c m i
· · · · (6)
′
AccordingtoBlume[26],themagneticcrosssectionfor +m e sin(q R +δ )
a a m i
· ·
polarized neutron is given as
Here, e , e , and e are the unit vectors along the a, b,
a b c
dσ and c-axis. Then, Eq. (3) can be written as [21, 26]
exp iQ(R R ) η η +iSˆ (η η )
(cid:16)dΩ(cid:17)∝X { i− j }(cid:2) j · i n j × i (cid:3)
i,j dσ m2+m2 2m m (Sˆ Qˆ)(Qˆ C) (7)
(2) (cid:16)dΩ(cid:17)± ∝h b c ± b· c· n· · i
4
The lastterm predicts the differentscatteringintensities both(dσ/dΩ) and(dσ/dΩ) aretheFouriercomponents
c s
for δ, and the relationcan be reversedby changingthe (symmetricandantisymmetric,respectively)ofthesame
sign±of either S or C. In fact, this behavior is clearly physical quantity σ+σ−. Therefore, the distribution of
n i j
observedin the results with E =+450kV/m(Figs. 3 (a) scattering intensities reflects the balance between sym-
c
and (b)). This means that Qˆ C is not zero, or in other metric and antisymmetric components of σ+σ− for the
i j
·
wordsthe magnetic structureofAF2 has the spiralcom- S=1/2case. Thismaybethecauseofthedeviationfrom
ponents in the bc-plane. Moreover, when the sign of E the Eq. (7). For example, in the extreme case of quan-
c
is reversed, the differential intensity relation is also re- tum fluctuation where the spins form the singlet state,
versed(Figs. 3(c) and(d)). This indicatesthatthe spin the commutation that σ+σ− = σ+σ− holds, there-
h i j i h j i i
chiralitydeterminesthedirectionofelectricpolarization. fore (dσ/dΩ)s = 0 and no differential intensity should
Conversely, the observed electric control of spin helicity be observed. The experimental observation of shrunk
directly proves that the ferroelectricity of LiCu O orig- magnetic moment [20] implies the large quantum fluctu-
2 2
inates from the transverse-spiral (cycloidal) spin struc- ation subsisting in the ordered spiral state. Therefore,
ture. Thus, the KNB model holds good even for the e - the quantum fluctuation of the vector spin chirality is
g
electron spin system, or under possibly large quantum likely to result in the reduced differential δ reflection
±
fluctuation inherent to the frustrated S=1/2 spins. The intensityofpolarizedneutrons,asobserved. Inaddition,
obtained geometric relation between spin chirality and several groups have implied that the magnetic structure
electric polarization is illustrated in Figs. 3 (e) and (f). of AF2 would be more complicated than the simple bc-
The sign of the coupling constant in Eq. (1) is negative spiral[15, 19]. This may also require some modification
(A < 0), which agrees with the theoretical prediction in Eq. (7). Note howeverthat evenwith any other mag-
[27]. Note that the sign of A[28] is different from the netic structure the observed difference for the opposite
caseof TbMnO3[21]. We alsomeasuredthe profileswith neutronspinsSn reflectsthechiralityinthebc-plane(see
E =0 andfound no difference for the intensity between Eq. (2)). For the thorough understanding, further anal-
c
δ reflections nor between the neutron spin states. This ysisofthemagneticstructureanditsquantumdynamics
±
should be due to the coexistence of opposite ferroelec- will be needed.
tric domains (or clockwise/counter-clockwise spin-spiral
In summary, the polarized neutron study was per-
domains) for the zero electric-field case. formedonthequantum-spinchainmagnetLiCu O . We
2 2
An unresolved problem at this stage is the ratio of confirmed the coupling between spin vector chirality of
scattering intensity between the stronger and weaker the transverse bc-spiral structure and the direction of
reflections. From Eq. (7), the elliptic ratio of electric polarization along the c-axis. This proves that
the spiral spin, mb/mc (or mc/mb), is estimated as even with the eg-electron system under the large quan-
(√ION √IOFF)/(√ION+√IOFF) forthecaseofclas- tum fluctuation the spin-current model or the inverse
| − |
sical spin[21]. On the basis of the data shown in Figs. Dzyaloshinskii-Moriya mechanism still works. The dif-
3 (a)-(d), this expression gives mc/mb (or mb/mc) = ferential intensity of polarized neutron reflections show
0.09 0.20. On the other hand, the aforementioned re- a clear deviation from that expected for the classical bc-
∼
sults on the Sn Q setup suggests the nearly equal value spiral spin structure, implying the importance of quan-
⊥
for mb and mc. As the origin of this discrepancy, the tum fluctuation in this S =1/2 helimagnet.
coexistence of different polarity domains might be sus-
The authors thank T. Arima, N. Furukawa, N. Na-
pected. However,we confirmed the saturationof electric
gaosa,S.Onoda,H.Katsura,J.Fujioka, Y.Shimada,H.
polarizationat E =350kV/m,withthesame(Al)elec-
| c| Sakai , S. Iguchi and Y. Onose for enlightening discus-
trode used in the neutron scattering study. Also on the
sions. This work was partly supportedby Grants-In-Aid
same sample, the Ag electrode was tested to confirmthe
for Scientific Research (Grant No. 16076205, 17340104,
identicalsaturationvalueofelectricpolarization. There-
19052002)from the MEXT of Japan.
fore,webelievethatthe singledomainstatewasrealized
in the S Q setup, and the above apparent discrepancy
n
k
should be ascribed to a more intrinsic origin. The mea-
suredtemperature (7K)mightnotbe lowenoughto sat-
uratethespinorder. However,theP valueat7Kalready [1] M. Fiebig, J. Phys. D : Appl.Phys. 38, R123 (2005).
reaches 80-90%of the 2K value (see Fig. 1 (d)); thermal [2] T. Kimuraet al.,Nature426, 55 (2003).
fluctuation alone is not enough to decrease the spin el- [3] Y. Tokura, Science 312, 1481 (2006).
lipticity m /m . One of other possibilities is the effect [4] S. -W.Cheong et al.,Nature Mater. 6, 13 (2007).
c b
[5] H. Katsura et al.,Phys. Rev.Lett. 95, 057205 (2005).
of quantum fluctuation. In case of S=1/2 quantum-spin
[6] T. Kimuraet al.,Phys.Rev. B 71 224425 (2005).
systems like LiCu O , the validity of the classical-spin
2 2 [7] T. Goto et al.,Phys.Rev. Lett.92 257201 (2004).
treatment as done in Eqs. (6) and (7) is no longer guar-
[8] G. Lawes et al., Phys.Rev.Lett. 95, 087205 (2005).
anteed. For a more rigorous argument, we have to go [9] Y. Yamasaki et al., Phys.Rev.Lett. 96, 207204 (2006).
back to Eqs. (3) - (5). According to these expressions, [10] K. Taniguchi et al., Phys.Rev.Lett. 97, 097203 (2006).
5
[11] T. Kimura et al.,Phys. Rev.B 73, 220401(R) (2006). [22] S. Zvyagin et al.,Phys.Rev.B 66, 064424 (2002).
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[14] S.Seki et al.,Phys. Rev.B 75, 100403(R) (2007). [25] Slightly different wavenumbers between ±δ is perhaps
[15] S.Park et al.,Phys. Rev.Lett. 98, 057601 (2007). due tothe misalignment of thesample.
[16] T. Masuda et al.,Phys.Rev. Lett.92, 177201 (2004). [26] M. Blume, Phys. Rev. 130, 1670 (1963).
[17] A.A.Gippiusetal.,Phys.Rev.B70,020406(R)(2004). [27] C. Jia et al.,Phys. Rev.B 76, 144424 (2007).
[18] T. Masuda et al.,Phys.Rev. B 72, 014405 (2005). [28] SignoftheSn-dependentterminEq.(7)iswronginref.
[19] A.Rusydiet al.,to be published. [21],becauseofthedifferentdefinitionofQfromref.[26].
[20] L. Capogna et al.,Phys. Rev.B 71, 140402(R) (2005). Inthecorrectdefinition,A>0isobtainedforTbMnO3.
[21] Y.Yamasaki et al.,Phys.Rev.Lett. 98, 147204 (2007).