Electric field control of chiral magnetic domains in the high-temperature multiferroic CuO

0.59 MB·English

by  P. Babkevich

Electric field control of chiral magnetic domains in the high-temperature multiferroic
CuO
P. Babkevich,1,∗ A. Poole,2 R. D. Johnson,1,3 B. Roessli,2 D. Prabhakaran,1 and A. T. Boothroyd1
1Department of Physics, Oxford University, Oxford, OX1 3PU, United Kingdom
2Laboratory for Neutron Scattering, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland
3ISIS facility, Rutherford Appleton Laboratory-STFC, Chilton, Didcot, OX11 0QX, United Kingdom
(Dated: January 17, 2012)
Wehavestudiedthehightemperaturemultiferroiccupricoxideusingpolarizedneutrondiffraction
as a function of temperature and applied electric field. We find that the chiral domain population
2
can be varied using an external electric field applied along the b axis. Using representation anal-
1
ysis we derive the incommensurate magnetic structure in the multiferroic phase. The origin of
0
themagnetoelectric coupling is consistent with models based on theinverseDzyaloshinskii–Moriya
2
interaction, but is different from thesimple cycloidal mechanism.

n
a PACSnumbers: 75.85.+t,75.25.-j,77.80.-e,61.05.F-
J

6
1 I. INTRODUCTION important role in the multiferroic mechanism.
Overall, it is widely accepted that the competing ex-

] changeinteractionstogetherwithgeometricalfrustration
l Materialsinwhichferroelectricandmagneticorderpa-
e lead to a non-collinear magnetic ordering which breaks
rameters couple are of great interest for application in
-
inversion symmetry and allows polar lattice distortions.
r spintronic devices. Such materials are termed multifer-
t What is different about CuO is that a strong Cu–O–Cu
s roic andhavebeenactivelystudiedinrecentyears.1–3 Of
. superexchange interaction along the [1,0,¯1] direction al-
t particular interest are the type-II multiferroics in which
a lows it to remain ordered at high temperatures.9,15–17
themagneticorderisdirectlyandstronglycoupledtothe
m
If the magnetoelectric coupling is strong enough, it
electricpolarization. Thestrongcouplingisoftenassoci-
- should be possible to vary the magnetic domain popu-
d ated with non-collinear magnetic structures which orig-
lation with an external electric field. The domain pop-
n inate from competition and frustration in the magnetic
ulation can be monitored by polarized neutron or pho-
o interactions. Most of the strongly-coupled multiferroics,
c therefore, exhibit their unusual properties at low tem- ton diffraction. This type of experiment has been per-
[ formed previously on several different multiferroics, and
peratures (< 40K). An exception is cupric oxide, which
1 develops a multiferroic phase below 230K.4 as well as demonstrating the manipulation of magnetic
domains by electric fields these experiments have pro-
v The magnetic phase diagramof CuO in zero magnetic
9 vided useful information about the magnetoelectric cou-
fieldcontainstwophases,AF1andAF2.5–9 Thelowtem-
0 pling mechanisms.18–23
peratureparaelectricphase(AF1)isasimplecommensu-
3 Here we present the results of a polarized neutron
3 rateantiferromagnetwhichisstableuptoTN1 =213.7K. diffraction study of the magnetic response of CuO in an
1. AtTN1 thereisafirst-ordertransitiontothemultiferroic appliedelectric field. We demonstrate thatthe magnetic
phase (AF2), which has a complex, non-collinear incom-
0 domains in CuO can be switched by an electric field ap-
mensurate magnetic structure and a ferroelectric polar-
2
plied along the ferroelectric polarization axis. The field
1 ization along the crystallographic b axis. Both magnetic
required to reverse the domain population is found to
: and ferroelectric orders vanish above T =230K.
v N2 be closely related to the temperature. However, it does
i The mechanism that causes the simultaneous emer- not appear to be possible to transform the crystal into a
X
genceofferroelectricityandmagneticorderinginCuOis
single magnetic domain even at large applied fields.
r still under discussion and a number of theories have re-
a
cently been proposed. A phenomenological Landau the-
ory approach was employed by Toledano et al. (Ref. 10)
II. EXPERIMENTAL PROCEDURE
to investigate the two ordered phases. Using this theory
they were able to explain the sequence of phases, with
the incommensurate multiferroic phase at a higher tem- A single crystal sample of CuO was grown from a
perature than the simple antiferromagnetic phase, and melt of high purity cupric oxide (99.995%) by the opti-
predictedthatbothphasetransitionsarefirstorder. Gio- calfloating-zonemethod.24 Thecrystal,whichwassingle
vannettietal.(Ref.11)andJinetal.(Ref.12)performed phaseanduntwinned,wascutwithparallelflatfacesper-
densityfunctionalcalculationswhichsuggestthatthefer- pendicularto the baxisgivingaplate-likesampleofsur-
roelectric polarizationis induced via the Dzyaloshinskii– face area of 21mm2, thickness 0.9mm and mass 0.175g.
Moriya(DM)interaction13,14betweenneighboringchains An8–10nmlayerofchromiumand40–50nmofgoldwas
ofnon-collinearspins. They further arguethatthe small evaporated onto the flat faces. Electrical contacts were
incommensurability in the AF2 phase does not play an attached using silver paste and the sample was sealed in
2
fromthesamerodastheneutrondiffractionsample. The
surface area of this sample was 10mm2 and its thick-
ness was 1.7mm. The sample was cooled from 250K in
the paraelectric phase to 220K in the multiferroic phase
with a potential difference of 180V applied parallel to
b. Upon reaching 220K, the electric field was turned off
andthesurfacechargeontheCr/Aucoatingwasallowed
todischarge. Thepyroelectriccurrentwasthenrecorded
either upon warming to above T or cooling down be-
N2
low T , using the same field-cool procedure before each
N1
measurement. The heating/cooling rate was 1Kmin−1.
To eliminate systematic errors, an applied electric field
parallel and antiparallel to b was employed.
Zero-field-cooled magnetization measurements were
performed with a SQUID-based magnetometer (Quan-
tum Design). The measuring field was 1000Oe. A slow
heating rate of 0.01Kmin−1 was used close to the sharp
T transition,andarateof0.1Kmin−1 wasusedabove
N1
and below it.
III. MAGNETIC REPRESENTATION
ANALYSIS
FIG. 1. (color online). The observed magnetic structures of
CuO in projection onto the a–c plane. The four Cu sites in
The antiferromagnetic structure of the AF1 phase
the conventional structural unit cell (indicated by the paral-
lelogram) are labelled 1–4. The colored circles represent the which exists below TN1 ≈ 213K is shown in Fig. 1(a).
componentofthemomentalongb: lightredisalong+b,dark The magnetic propagation wavevector is qcm =
blue is along −b, and blue circles with arrows are moments (0.5,0,−0.5),andthespinslieparalleltothebaxis. The
which lie in the a–c plane. Panel (a) shows the commensu- multiferroic phase (AF2) exists within the narrow tem-
ratemagneticstructureintheAF1phasewhosemomentsare peraturerangebetweenT andT . Neutrondiffraction
N1 N2
all parallel or antiparallel to b. Panel (b) shows the spin ar- experiments have shown that the magnetic structure in
rangement in theincommensurateAF2phase. Themoments
this phase has anincommensuratepropagationwavevec-
lie in the planes indicated by thedotted lines, which contain tor q =(0.506,0,−0.483). The Cu2+ moments rotate
thebaxisandthedirectionv. TheAF2structurehastwodo- icm
withanapproximatelycircularenvelopeintheplanecon-
mains of opposite handedness. Only the left-handed domain taining the vectors b and v = 0.506a∗+1.517c∗, where
is shown here.
a∗ and c∗ are reciprocal lattice basis vectors.7,8 Spon-
taneous ferroelectric polarization is found to develop in
an aluminium vacuum can to avoid electric field break- the AF2 phase along the b axis.4 No ferroelectricity is
down. observed for T <TN1 or T >TN2.
Polarized neutron diffraction measurements were per- The crystal structure of CuO can be described by the
◦
formedonthe TASP/MuPADinstrumentatSINQ,Paul monoclinic C2/c space group, with β = 99.55 and lat-
Scherrer Institut.25,26 The sample was aligned with the tice parameters of a=4.69˚A, b=3.42˚A and c=5.13˚A
a and c axes in the horizontalscattering plane such that at 300K.27 For both magnetic phases, the magnetic
theappliedelectricfieldwasalongthe−bdirection. The propagation vector q is left unchanged by the identity
scattering vector is defined as Q = ki − kf where ki g1 = {E|0,0,0} and glide plane g2 = {σy|0,0,0.5} sym-
and k are the incident and scattered neutron wavevec- metry operations of the paramagnetic space group. The
f
tors, and ki = kf = 1.97˚A−1. The flipping ratio, as little groupGq is therefore{g1,g2}. The irreduciblerep-
measuredonthe(002)and(200)structuralBraggpeaks, resentations of Gq are Γ1 and Γ2, whose characters are
was found to be 17. The experimental data presented given in Table I. The primitive cell of CuO contains two
here have not been corrected for the non-ideal neutron Cuatoms,buttheC-centering{E|0.5,0.5,0}thengener-
polarization, but corrections have been included in all atestwomoreCusitessuchthattherearefourCuatoms
calculations from magnetic structure models. The back- in the conventionalunit cell.
ground signal in the different polarization channels was The components of the two Cu spins in the prim-
measured at positions away from the Bragg peaks and itive cell form a 6-dimensional representation Γ of
mag
found to be negligible. G which reduces to Γ = 3Γ + 3Γ . Symme-
q mag 1 2
Pyroelectric current measurements to obtain the elec- try analysis28 fixes the relative phases between Cu sites
tric polarization of CuO were made with a customized and the experimentally-determined spin directions fur-
PPMS (Quantum Design) insert. The crystal was cut ther constrain the magnetic structures. The magnetic
3
Gq {E|0,0,0} {σy|0,0,0.5} differentsymmetryforthe bandv magneticcomponents
Γ1 1 ϕ (Mb(1)Mv(2) or Mb(2)Mv(1)) do havea net electric polariza-
Γ 1 −ϕ tion in the spin current model.
2
OfthefourassumedAF2magneticstructuresforCuO,
TABLE I. Character table of the irreducible representations only M(1)M(2) is consistent with neutron diffraction ex-
b v
of Gq with thephase factor given as ϕ=−exp(iπqz). periments. TheM(2)M(1) structurecanbe ruledoutbe-
b v
causeitpredictsaverysmallstructurefactor(identically
j Mb Mv zero for an isotropic Cu form factor) for the Q = qicm
Γ1 1 cos(q·t+θb) cos(q·t+θv) fundamental diffraction peak, while the Mb(1)Mv(1) and
2 cos(q·t+qcπ+θb) −cos(q·t+qcπ+θv) Mb(2)Mv(2) structuresgiveverypooragreementwithneu-
Γ 1 cos(q·t+θb) cos(q·t+θv) tron polarimetry measurements.29 The Mb(1)Mv(2) struc-
2 2 −cos(q·t+qcπ+θb) cos(q·t+qcπ+θv) ture is shown in Fig. 1(b), and has been used in all sub-
sequent analysis presented here.
TABLE II. The symmetry-allowed magnetic structures By enumerating the possible symmetry-allowed mag-
of CuO. The AF2 structure has ordering vector q = neticstructuresfortheAF1andAF2phasesofCuOand
(0.506,0,−0.483) and can be resolved into components par- comparing these with experiment, we conclude that the
allel to b and v = 0.506a∗ +1.517c∗. The AF1 structure structures shown in Fig. 1 are the correct ones. We can
has q = (0.5,0,−0.5) and only has components along the b
therefore use these to quantify changes in domain pop-
axis. The Cu site positions j are: (1) 1,1,0; (2) 1,3,1; (3)
4 4 4 4 2 ulation induced by an applied electric field. In the next
3,3,0; (4) 3,1,1, as indicated in Fig. 1. The lattice trans-
4 4 4 4 2 sectionwedescribebrieflythemethodologyofourexper-
lation vectors are denoted by t, and the moments of atoms
iment.
on sites 3 and 4 are found using the C-centering translation
tC =(1/2,1/2,0), resulting in an additional phase of q·tC.
For the AF1 structure, θ =π/4, and for the AF2 structure,
b
IV. SPHERICAL NEUTRON POLARIMETRY
θb =0 and θv =π/2.
In general, magnetic scattering of neutrons involves a
components of the structures consistent with this analy- change in the polarization state of the neutrons. Spher-
sis are given in Table II. ical neutron polarimetry is a technique which measures
the polarization of the scattered neutron beam for an
The commensurate AF1 phase can be described us-
arbitraryincidentbeampolarization. Theresultsareex-
ing just M , the basis vector along b. The spin struc-
b
pressed in terms of a polarization matrix,
tures corresponding to the pure Γ and Γ irreducible
1 2
representations, Mb(1) and Mb(2), are given in Table II. P = Iαβ −Iαβ, (1)
The choice of phase factor θb = π/4 fixes the amplitude αβ Iαβ +Iαβ
of the magnetic moment on each Cu site to be equal.7
where α and β indicate the polarization of the incident
Single-crystal neutron diffraction measurements of mag-
and final beam, respectively. I is the magnetic Bragg
netic Bragg peak intensities below T have established αβ
N1
peak intensity, given by
that the structure for the AF1 phase has the Γ symme-
1
try. Iαβ =|hα|M⊥(Q)·σ|βi|2, (2)
The incommensurateAF2 phase has magneticcompo- where σ is the Pauli operator for the neutron spin and
nents alongboththe b andv directions, M andM ,re-
b v M⊥(Q) is the component of M(Q) perpendicular to Q,
spectively. To correctly describe the AF2 spin structure
where M(Q) is the Fourier transform of the magnetiza-
we set θ =0 and θ =π/2 so that the spins rotate with
b v tion.
a circular envelope, as observed experimentally. Assum-
For CuO, M(Q) may be written,
ing the M and M components can each be described
b v
by a single irreducible representation, Γ1 or Γ2, there M(Q)=XXfj(Q)Mj(t)eiQ·(t+rj), (3)
are four possible magnetic structures which we denote
t j
by M(n)M(m). These structures are chiral, so each has
b v wherer isthe displacementofthe jthCusite(j =1–4)
j
two domains of opposite handedness related by spatial
fromtheoriginofthe conventionalunitcell, f (Q)isthe
j
inversion. When the b and v magnetic components be-
Cu magnetic form factor, and t is a lattice translation
(1) (1)
longtothesameirreduciblerepresentation,i.e.Mb Mv vector. The components of Mj(t) are given in Table II.
or M(2)M(2), this corresponds to a spin arrangement in In an ideal measurement with perfect neutron polar-
b v
which the spins on sites 1 and 3 rotate in the opposite ization the polarization matrix for CuO is given by
sense along the a axis to those in sites 2 and 4. This
−1 0 0
results in zero net electric polarization in the unit cell  
assuming the magnetoelectric coupling depends on the P= C −D 0 , (4)
 
spin current S1×S2. The two magnetic structures with C 0 D
4
where,
2ℑ{M∗M } |M |2−|M |2 −3 3 a) Hkb∗
C = z y , D = z y . (5) 10
|M⊥|2 |M⊥|2 × Hka∗
)
1
Here we have used the Blume coordinate system,30,31 −ol 2.5
with the x axis parallel to Q, the z axis perpendicu- 3m Hkc∗
m
lar to the scattering plane, and the y axis chosen to c
(
make up a right-handed set of axes. In this axis sys- χ 2
tem M⊥ = (0,My,Mz). The existence of a non-zero
‘chiral’ term C in the polarization matrix implies a non- 100 b)
collinear magnetic structure that is, by definition, con- 2) 50
−
structedfromorthogonal,out-of-phasecomponents,such m AF1 AF2 PM
as a helix or cycloid. Therefore, C = 0 in the collinear C 0
µ
(
AF1 structure of CuO, and in the AF2 phase, left- and Pe −50 Ekb∗
right-handed chiral domains will give C < 0 and C > 0,
respectively. Equivalently,ifwemeasuretheP andP −100
yx zx 300
)
elements on a magnetic reflection we can determine the −1 c)
m
relative proportions of the two chiral domains. m
TheparametersC andDcanbecalculatedfromEqs.5 V 200
for a general helicoidal magnetic structure. For the par- eld( Ekb∗
ticularcasewhenthespinsrotatewithacircularenvelope fi 100
e
v
in a plane perpendicular to q, the values for the reflec- ci
er
tion Q=q are C =±1 and D =0. In the AF2 phase of o
C 0

CuO, the propagation vector q is not perpendicular 205 210 215 220 225 230 235 240
icm
tothe planeofrotationofthe spins,butratherliesatan T(K)
angle of 107◦ to it — Fig. 1(b). For Q = q this results
in a small but non-zero value of D, and a magnitude of FIG. 2. (color online). Magnetic and dielectric properties
C whichis slightly less than1 for a single chiraldomain. of CuO. The upper panel (a) The zero-field-cooled magnetic
susceptibility measured with a field of H = 1000Oe applied
The polarization matrix measured on a nuclear reflec-
tion contains only the diagonal elements P = δ . parallel to the {a∗,b∗,c∗} reciprocal lattice vectors. (b)
αβ αβ The electric polarization with the electric field applied along
Measurements on nuclear (or equivalently magnetic) re- b∗ direction. Measurements were made with a poling field
flections are useful in estimating the amount of depolar- of 106Vmm−1 applied in opposite directions (red and blue
ization of the incident neutron beam due to stray fields. curves). (c) shows the temperature dependence of the coer-
This is quantified by a flipping ratio, defined for a nu- civefield measured byneutron polarimetry.
clear Bragg peak as the ratio of the scattering inten-
sity measured in the non-spin-flip to spin-flip channel,
R=Iαα/Iαα¯. (Ref.9). Thedifferentbehaviorsmayrelatetoslightdif-
ferencesinthechemicalcompositionoftheCuOcrystals.
TheelectricpolarizationP along±b∗ obtainedbyin-
e
V. EXPERIMENTAL RESULTS tegrating the pyroelectric current with a poling field of
106Vmm−1 is shown in Fig. 2(b). A sharp depolariza-
Temperature profiles of magnetic and dielectric prop- tion is observedclose to 213K coincident with the AF1–
erties of our CuO crystal are presented in Fig. 2. Fig- AF2 magnetic transition. A maximum polarization of
ure 2(a) shows temperature sweeps of the magnetic sus- approximately 100µCm−2 is recorded.
ceptibility recorded along the reciprocal lattice axes a∗, As discussed above, neutron polarimetry can be used
b∗ and c∗. A sharp discontinuous transition is observed to determine the relative proportions of the two chiral
along all directions at T = 213.7K. Above T the domains in CuO. Any changes in population induced by
N1 N1
susceptibility increases linearly with temperature up to an applied electric field will affect the P and P ele-
yx zx
T ≈230Katwhichpointachangeinslopeisobserved. ments in the polarization matrix while leaving the other
N2
Ourdataareinverygoodagreementwithpreviousmea- elements unchanged.
surements on CuO.4 Susceptibility measurements were CompletepolarizationmatricesP measuredwithan
αβ
made on cooling and warming. We find no hysteresis applied external electric field of ±670Vmm−1, recorded
at T to within 0.2K, despite the first-order nature of at 220K, are shown in Figs. 3(a) and (b). The data
N1
this transition. We observeda similarlyabruptandnon- clearly shows a reversal in the sign of the terms P
yx
hysteretic transition at T in magnetic neutron diffrac- and P . A decrease in the xx component in Fig. 3(c)
N1 zx
tion measurements also made on cooling and warming. is anomalous. Its origin is unclear but may be due to
Thisisatoddswiththelargehysteresisofapproximately a small misalignment of the crystal with respect to the
5K reported in diffraction measurements by Yang et al. Blume frameofreferencewhichalsocausesthe P term
xz
5
to be non-zero. 0.5 a) Ekb 0.6 c)
Measurements of the P component of the polariza-
yx 0
tion matrix ata constanttemperature of 220K after ini- Pαβ 0.3 ↑
tial field cooling are shown in Fig. 3(c). We find that −0.5 Ek−b
by sweeping the electric field, which is applied in the - D1=80%,D2=20%
b∗direction,from670Vmm−1to−670Vmm−1andback 0−.51 b) Ek−b Pyx 0
to 670Vmm−1 we observe a change in sign and magni-
tudeofPyx. Itfollowsthattheelectricfieldmustbecou- Pαβ 0 −0.3 ↓Ekb 220K
pled to the chiral magnetic domains. The coercive field, −0.5
D1=24%,D2=76%
defined as the electric field at which the chiral domains −1 −0.6
are equally populated, is approximately 90Vmm−1 at xx yx zx xy yy zy xz yz zz −500 E(Vm0m−1) 500
this temperature.
WemeasuredPyx loopsasafunctionofelectricfieldat 150 d) IIyyxx¯ e) IIyyxx¯
a series of temperatures within the AF2 phase. The co- b.) Ekb Ek−b
ercivefieldobtainedfromthese loops,showninFig 2(c), (ar100
are found to increase with decreasing temperature. The nts
u
systembecomessofterathighertemperaturesandthere- Co 50
foretheelectricfieldrequiredtobalancethedomainpop-
0
ulation is reduced up to the point of the phase transi-
−35 −34 −33 −32 −31 −30−35 −34 −33 −32 −31 −30
tion at 230K. No hysteresis and hence no coercive field ω(deg) ω(deg)
is found in the AF1 or paramagnetic (PM) phases. To
verify that the electric field switching of domains exists FIG. 3. (color online). Spherical neutron polarimetry data
only in the AF2 phase, hysteresis loops were also made for CuO with an applied electric field. (a) and (b) represent
at 240K (centered on q ) and 210K (centered on q thecompletepolarization matrix atqicm =(0.506,0,−0.483)
icm icm with applied field E of −670Vmm−1 and +670Vmm−1, re-
andq ). Thesemeasurementsshowedfield-independent
behavcimor of P , as expected. spectively. The color bars are the measurements, and the
We note thyaxt the magnitude of P never reaches the symbols(◦)arecalculatedassumingtheMb(1)Mv(2) AF2mag-
yx neticstructureandtakingintoaccountthenon-idealneutron
maximal value of close to ±1, as expected for the case
beam polarization. The fitted populations of the two chiral
of a single chiral domain in Eq. 5. Even at large electric magnetic domains are indicated. Panel (c) shows the Pyx
fields a significant depolarization is observed such that component of thepolarization matrix duringan electric field
a saturation value of |Pyx| ≈ 0.5, is reached at around sweepatafixedtemperatureof220K.Panels(d)and(e)show
±500Vmm−1 whenmeasuredat 220K.Hysteresisloops rockingscansthroughtheqicm magneticBraggpeakasmea-
measured at both lower and higher temperatures within sured in the yx and yx polarization channels. Measurements
theAF2phaseshowsimilarbehaviorwiththesaturation were made at constant applied electric field of −670Vmm−1
value of |P |≈0.5. (paneld) and +670Vmm−1 (panel e).
yx
To examine this effect further we performed rocking
scans through the q Bragg peak. Figures 3(d) and
icm
(e) show the intensities in the spin-flip and non-spin-flip anymagneticstructure. This sumruleisnotsatisfiedby
yxpolarizationchannelsatthe extremaofthe hysteresis our polarimetry data — see Figs. 3(a) and (b). We con-
loop. We would expect the left-handed helical domain clude, therefore, that electric field conversion to a single
(i.e. that shown in Fig. 1(b)) to scatter mainly into the magnetic domain is never achieved in CuO.
yxpolarizationchannel,andtheright-handedhelicaldo- Toquantify the domainpopulations,we haveusedthe
main to scatter mainly into the yx¯ channel. Taking into modelfortheAF2magneticstructurewherebasisvectors
account the non-ideal neutron beam polarization in the alongbcorrespondtotheΓ representationandthebasis
1
experiment we calculate the ratio I /I to be 15 for vectorsina-c planeto Γ . Acorrectionwasmade to the
yx yx¯ 2
a single magnetic domain. However, from Fig. 3(d) this simulations to account for non-ideal beam polarization
ratio is closer to 3.4. assuming a beam polarization efficiency of 94%. We fit-
This discrepancy suggests either that the assumed ted the complete measured polarization matrices allow-
magneticstructurefortheAF2phaseofCuOisincorrect, ing just the domain fraction to vary. The domains D1
orthat complete switching betweenmagnetic domains is and D2 correspond to left- and right-handed spin struc-
notachievedeventhoughP apparentlysaturatesatthe tures, respectively. The electric field switches between
yx
highest measured electric fields. The magnetic structure the domains, with a large positive electric field yielding
intheAF2phaseofCuOhasbeentightlyconstrainedby a predominantly left-handed domain and, conversely, a
previous neutron diffraction and polarimetry studies,7,8 large negative field promoting the right-handed domain.
as well as analysis of our own results,29 and so it seems We find that typically, in the AF2 phase, at electric
reasonable to assume that the magnetic structure that fields approaching saturation in P , the magnetic do-
yx
we consider is indeed the correct one. Moreover, it can mains are populated in approximately 80:20 proportion.
beshownthatP2 +P2 +P2 =1forasingledomainof We have observed, therefore, that the electric field does
yx yy yz
6
not induce a single magnetic domain. Possible reasons tionaxis. Thiscouplingislimitedtoasmallclassofcrys-
forthisare,(i)crystaldefects actingtopinthe magnetic talstructureswhicharetermedferroaxial.32 CuOadopts
structure and preventfull domain reversal,(ii) the high- thespacegroupC2/c,amemberoftheferroaxialcrystal
temperature of the multiferroic phase causing thermal class, with a two-fold rotation axis parallel to the crys-
relaxationofthe domainpopulationinthe time frameof tallographic b-axis. Coupling between magnetic chiral-
the experiment, or (iii) a reduced electric field strength ity and the crystal structure could therefore result in an
near the edges of the crystal caused by the incomplete electricpolarizationparalleltotheb-axis,asobserved. It
coating of the surfaces with the gold electrodes. would be of interest to include this chiral term in future
DFT calculations.
A recent study of the bulk magnetoelectric properties
VI. DISCUSSION
of polycrystalline CuO showed that the application of
a magnetic field has little effect on the ferroelectricity
Density functional theory calculations of the electric and concluded that the magnetoelectric coupling is very
polarization in the multiferroic phase of CuO were re- weak.33 Our method of applying an electric field and
portedinRefs.11and12. ThemagnitudeofPepredicted measuring the effect on the magnetic structure shows
by both groups is consistent with the experimental bulk that the magnetoelectric coupling is in fact strong. The
valueofapproximately100µCm−2. However,weobserve difference between these experiments is that in our work
that the direction of Pe is along +b for the left-handed weareprobingthecouplingbetweenthemacroscopicpo-
domain, in contradiction to the direction predicted by larizationandaspatially-varyingmagnetization,whereas
one of the models.12 inRef.33thecouplingisbetweenthemacroscopicpolar-
The magnetically-induced ferroelectric polarization in ization and a uniform magnetization. These studies are
CuO has been explained through a number of models, therefore complementary, not contradictory.
all of which are based upon the inverse Dzyaloshinkii–
Moriya interaction between nearest-neighbor, non-
collinear spins. For such spin structures, the DM inter-
action may reduce the system’s energy, as described by
VII. SUMMARY
the following term in the Hamiltonian,
H = D ·(S ×S ). (6) We have used polarized neutron diffraction to show
DM X ij i j
that an electric field applied along the b axis is able
i,j
to switch between magnetic domains in CuO. The re-
The DM vector, D , can be resolved into components sults demonstrate that the ferroelectric polarization is
ij
perpendicular and parallel to the vector r connecting directly coupled to the chiral magnetic order. We have
ij
the spins as, solved the magnetic structure in the multiferroic phase
by representation analysis and confirmed that it is con-
Dij =Pij ×rij +σijrij, (7) sistent with previous data and that it supports a ferro-
electricpolarization. Themicroscopicoriginofthe ferro-
whereP andσ areapolarvectorandpseudoscalar,re-
ij ij electricity is consistentwith models basedon the inverse
spectively. InCuO,Pij isproportionaltoalocalelectric Dzyaloshinkii-Moriyamechanism.10–12
polarization,and σ is related to the magnetic chirality.
ij
Usuallyonlythefirsttermisconsideredwhenexplaining
the evolution of ferroelectricity, as it explicitly includes
a polar vector. This is indeed the case in all previous
ACKNOWLEDGMENTS
studies of CuO. Kimura et al. (Ref. 4) consider a spin
cycloid whose propagation vector is along the direction
ofthe incommensurate modulationq −q , andGio- We wish to acknowledge M. Zolliker, M. Bartkowiak,
icm cm
vanetti et al. (Ref. 11) and Jin et al. (Ref. 12) find that R.Makin,Y.BodenthinandU.Staubforimportantcon-
theimportantmagnetoelectricinteractionisbetweenap- tributions relating to the technical aspects of the exper-
proximatelyperpendicularspinsonneighboringchainsof iment. We are also grateful to I. Cabrera, J. Lorenzana,
Cu atoms running along the [1,0,1] direction and does P. G. Radaelli and T. Kimura for insightful discussions.
not depend on the incommensurate modulation. PBisgratefulfortheprovisionofastudentshipfromthe
The second term ofEq. 7 does not inherently give rise UK Engineering and Physical Sciences Research Coun-
to a polarization. However, it has recently been shown cil and Paul Scherrer Institut. This research project has
that the chirality of a magnetic structure, σ , can in- been supported by the European Commission under the
ij
duceanelectricpolarizationthroughcouplingtoaunique 7thFrameworkProgrammethroughthe‘ResearchInfras-
structural rotation.32 Furthermore, the electric polariza- tructures’actionofthe‘Capacities’Programme,contract
tionisthenstrictlyconstrainedtolieparalleltotherota- No: CP-CSA INFRA-2008-1.1.1Number 226507-NMI3.
7
∗
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