Table Of ContentIdentification and mechanical control of ferroelastic domain structure in
rhombohedral CaMn O
7 12
Renliang Yuan,1 Lian Duan,1 Xinyu Du,1 and Yuan Li1,2,∗
1International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
2Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
We report on observation of ferroelastic domain structure in single crystals of multiferroic
CaMn7O12 at room temperature. Two types of ferroelastic domain wall are found, consistent with
the material’s rhombohedral symmetry that is reduced from cubic symmetry at higher tempera-
tures. UsingRamanspectroscopyalongwithothermeasurements,wedevelopasystematicmethod
5
to determine the microscopic domain orientation. Moreover, we find a switching behavior of the
1
domains, which allows us to detwin the crystals conveniently at room temperature using a mod-
0
erate uniaxial compression. Our result paves the way for further spectroscopic study and domain
2
engineering in CaMn7O12.
n
a PACSnumbers: 75.85.+t,77.80.Dj,78.30.-j
J
4
I. INTRODUCTION cell below Ts, along with the shortened body diagonal,
] are denoted by yellow/blue cubes and magenta lines, re-
l
e spectively, in Figs. 1(a-b), omitting most details.
Ferroic (ferroelectric, ferromagnetic, ferroelastic) do-
- Since current methods of growing single crystals of
r main walls have aroused persistent research interest due
t CaMn O all take place at temperatures far above the
s to their significance in both fundamental research and 7 12
t. promising applications.1–3 In so-called multiferroic ma- ferroelastic structural transition temperature Ts,15,18,19
a one can reasonably expect ferroelastic domains to form
terials, different ferroic order parameters coexist and ex-
m upon cooling the crystals to room temperature, as a
hibit mutual coupling, hence allowing for the manipula-
- result of simultaneous nucleation from different parts
d tion of one ferroic property through another. This co- in the crystal, just like those in YBa Cu O 23 and
n existence gives rise to composite domain walls4–6 that 2 3 6+δ
BaFe As .24 In this paper, we focus on domain struc-
o might be a key to utilizing the mutual controllability of 2 2
turesinCaMn O atroomtemperature,wherethecom-
c ferroic properties in applications. Furthermore, domain 7 12
[ poundisrhombohedralbutnotferroelectric. We present
walls can exhibit distinctly different properties from the
1 bulk,7–9 leading to the possibility of using the domain directobservationsoftwotypes ofstripe-like ferroelastic
domainstructures in single crystals. By using polarized-
v walls as device. In addition to the intensively studied
lightmicroscopy,Ramanspectroscopyandstylussurface
7 systemBiFeO ,10intriguingdomainstructureshavebeen
6 observed and3tuned in manganites.4,6,11–13 profiler, we study the ferroelastic domain structures we
6 observe. We propose a method to uniquely determine
0 CaMn7O12 is a “type-II” multiferroic material14 with surface domain structures using Raman spectroscopy in
0 very large ferroelectric polarization induced by mag- combinationwithstylusprofiler. Moreover,weidentifya
1. netic order, and has been the subject of considerable switchingbehaviorofthe ferroelasticdomains,andshow
0 recent research efforts.15–19 Unlike in many widely stud- that the crystals can be mechanically detwinned by a
5 ied ferroelectric materials, the occurrence of ferroelec- moderate uniaxial compression. These findings are im-
1 tricity in CaMn7O12 is preceded by a ferroelastic struc- portant for further spectroscopic studies of CaMn7O12
v: tural phase transition at higher temperatures, making that require single-domain samples, and may facilitate
i it an ideal platform to study the ferroelastic and ferro- future investigationsof domain and domain-wallproper-
X
electric domain structures separately. At high tempera- ties in multifunctional oxides.
r tures, CaMn O possesses the AC B O cubic struc-
a 7 12 3 4 12
ture which is a derivative of simple perovskite ABO .20
3
Upon cooling, a first-order phase transition occurs at
T ≈ 440 K. The four Mn3.25+ ions in each formula unit II. EXPERIMENTAL METHODS
s
are charge-orderedinto three Mn3+ and one Mn4+ ions.
The body diagonal of the high-temperature cubic cell High-qualitycube-shapedsinglecrystalsofCaMn O
7 12
thatruns throughthe Mn4+ ions shrinksalittle bit, and were grown with a flux-reaction method15 at a cooling
◦
becomes the c-axis of the new rhombohedral unit cell in rate of 5 C/h. Natural facets of the crystals are par-
hexagonal basis.21,22 This hexagonal c-axis plays an im- allel to crystallographic {100} planes, where the sub-
c
portantrolethatnotonlydeterminesthedirectionofthe script “c” denotes pseudo-cubic notation. The samples
incommensurateorbitalorderestablishedbelowT =250 were characterizedas described elsewhere.19 Optical im-
o
K,16 butalsosetsthedirectionofthegiantimproperfer- ages ofcrystalsurfaces were taken with a polarized-light
roelectric polarization that arises from the helical mag- microscope Olympus BX51 with polarizer and analyzer
netic order below T = 90 K.15,17 The pseudo-cubic setinanalmostperpendicularconfiguration. Differential
N1
2
interference contrast apparatus was installed to enhance (a) (b)
image contrast. Surface profiles of crystal facets were
measuredwitha KLA-TencorP-6StylusProfilerusinga
contactforceof0.5mg. Ramanscatteringmeasurements
were performed in a back-scattering confocal geometry
using the 632.8 nm line of a He-Ne laser for excitation.
z
The diameter of the focused laser spot is estimated to
be less than 5 microns. A Horiba Jobin Yvon LabRAM
y
HREvolutionspectrometer,equippedwitha600gr/mm x
grating and a liquid-nitrogen-cooled CCD detector, was
(c) (d)
used to analyze the Raman spectra.
III. RESULTS AND DISCUSSIONS
A. Possible domain-wall orientations
Figures 1(c-d) display typical polarized-light optical
(e) (f)
images of multi-domain samples. Ferroelastic domains
manifest themselves in a regular bright-and-dark stripe A C
pattern. Whileonlythecrystals’topfacesareshown,the
domainstructuresactuallyextendtospantheentirecrys-
B D
tals,i.e.,aconsistentpatternisfoundonthesidefacesas A D
well. This is illustrated(with exaggeratedrhombohedral
distortion of the domains) in Fig. 1(e) and Fig. 1(f), in
which the slab-like domains are found to stack along the
pseudo-cubic h100i and h110i directions, respectively.
c c
Onslightly twinned samples,especially those grownata
◦ FIG. 1. (Color online) (a), (b) Schematics of {100}c and
coolingrateslowerthan5 C/h,wecanoccasionallyfind
{110}c typedomainwallsinCaMn7O12. Theblueandyellow
stripes near the edges of the faces that do not span the
cubesindicatedifferentdomainswith themagentalinesindi-
entire crystal.
cating the shortened body diagonals. Domain walls are de-
When the crystals are heated to temperatures above noted by theplanes separating the cubes. (c), (d) Polarized-
Ts, the stripe patterns gradually disappear, consistent lightopticalimagesofsinglecrystalswith{100}c and{110}c
with the recovery of a single-domain cubic structure domainwalls. Inadditiontothedomainstructureswhichap-
above T . Cycling the temperature through T can each pear as regular bright-and-dark stripe patterns, growth ter-
s s
timeleadtoacompletelydifferentstripepatterninrhom- racesareseenespeciallyin(c),buttheydonotseemtoaffect
the domain distribution. (e), (f) Three-dimensional illustra-
bohedral phase, which indicates that the domain forma-
tions of the crystals in (c) and (d), respectively, with blue
tion is not pinned by disorder or defects, and it in turn
and yellow slabs indicating the two different domains. Sur-
confirms the high quality of our samples. On the other
faces with four inequivalent domain structures are labeled as
hand, when the samples are cooled down to cryogenic
A,B, C, and D.
temperatures,thestripepatternremainsevenbelowT
N1
= 90 K in the ferroelectric phase. Hence it is possible to
control low-temperature ferroelectric domain structures
Domain-wall orientations in ferroelastic materials can
by pre-setting a desired ferroelastic domain structure in
be understood by the equilibrium boundary condition
the paraelectric phase.
(strain compatibility), which can be written as the
Grown at temperatures far above Ts and then cooled following:25
to roomtemperature,mostcrystalscontainmultiple fer-
roelasticdomainsduetosimultaneousrhombohedraldis- 3
′
tortions that nucleates from different parts of the sam- [S −S ]x x =0, (1)
X ij ij i j
ples. Even though as-grown crystals can occasionally i,j=1
be foundinasingle-domainstate,15,16 furthermanipula-
′
tions may affect the domain structure. For example, to where S and S are the strain tensors of two adjacent
acquirepureA RamanspectrainCaMn O ,theprepa- domains. Indices i and j = 1, 2, and 3 denote Cartesian
g 7 12
rationof apolished surfaceis required,19 andthe polish- coordinates, and all possible (x1,x2,x3) that satisfy the
ing process will inevitably exert mechanical stress onto equation constitute permissible boundaries between the
the sample, thus raising the risk of twinning it. We will two domains.
discuss domain-switching behavior under external forces The rhombohedral phase of CaMn O belongs to
7 12
and how to utilize it to detwin crystals later. the ferroelastic species m3F3 with four possible domain
3
variants,26 whichistheresultofsymmetryloweringfrom asingle-domaincrystal. Here,XX denotesthatboth the
Im3 to R3.21 The four different spontaneous strain ten- incoming- and scattered-photon polarizations are along
sors can be written in the form:27 the [110] direction, whereas YY denotes that both po-
c
larizations are along the [110] direction; the hexagonal
0 d d 0 −d −d c
c-axis of the crystal is determined to be along the [111]
S = d 0 d , S = −d 0 d , c
1 2
direction (see below). The measurement configurations
d d 0 −d d 0
(2) are illustrated in upper-left inset of Fig. 2. Comparing
0 −d d 0 d −d the XX and YY spectra, one sees a clear difference in
S3 = −d 0 −d,S4 = d 0 −d, the 380 to 500 cm−1 region, where four peaks can be
d −d 0 −d −d 0 attributed to Ag and Eg modes.19 The difference is due
tothefactthatboththeincoming-andscattered-photon
with the corresponding shortened body diagonals lying
polarizationshaveanon-zeroprojectionalongthehexag-
indirection[111] ,[111] ,[111] ,and[111] ,respectively.
c c c c onalc-axisintheXXscatteringgeometry. Thisnon-zero
When any two of these four domain variants meet, the
projection leads to a larger (smaller) weight of the A
solutions to Eq. 1 correspond to one of the equivalent g
(E ) signals in the Raman spectrum compared to that
crystallographic planes {100} and {110} ,25 which is in g
c c obtained in the YY scattering geometry, in which the
perfect agreement with our observations in Fig. 1. The
photon polarizations are perpendicular to the hexagonal
two different types of domain walls are illustrated in
c-axis.
Figs. 1(a-b). Similar domain structures have been found
in BiFeO ,28 BaTiO ,29 LaAlO ,30 etc. All of these per- ByperformingbothXXandYYmeasurementsononly
3 3 3
one face of the cube and after obtaining the data shown
ovskiteshaverhombohedrallydistortedphase,the strain
tensors of which are in the same form27 as in our case. in Fig. 2, we can only tell that the hexagonal c-axis is
along either the [111] or the [111] direction. In order
Additionaldomainvariantsmayexistwhenferroelectric- c c
to unambiguously determine the hexagonal c-axis, one
ity sets in, but the number of ferroelastic variants will
◦ needs to perform additional measurements on adjacent
remain to be four since 180 ferroelectric domain walls
are not ferroelastic.25,31 faces of the cube. To verify the validity of our method,
wehaveperformedsixpairsofXXandYYmeasurements
There are a total of four inequivalent surface domain
on all faces of severalsingle-domaincrystals,all yielding
structures on any {100} face of a cube-shaped crystal.
c
consistentresultssupportingtheaforementionedpicture.
TheyarelabeledasA,B,C,andDintheillustrationsin
This gives us confidence on how to identify and orient
Figs.1(e-f). OnanA-typefaceonefindsh100i stripesin
c
single-domain crystals. Moreover, it suggests the possi-
the optical image, which continue onto adjacent faces as
bilitytodeterminetheorientationsofindividualdomains
h100i stripes;aB-typefaceexhibitsnostripessinceitis
c
in multi-domain samples using Raman spectroscopy.
single-domain;aC-typefacehash110i stripes; aD-type
c
facehash100i stripes,butunliketheA-type,thestripes For simplicity, here we use the relative inten-
c
continueontoadjacentfacesash110i stripes. Therefore, sity ratio between 390 and 425 cm−1, R =
c
for a given crystal with sufficiently large single-domain I(390cm−1)/I(425cm−1), to represent the key charac-
volumes that span the entire crystal, one can obtain a teristics of the XX and YY Raman spectra in Figs. 2.
rough idea about its domain structure by visually in- As can be seen from the upper-right inset, R captures
specting allofits sixfaces under a polarized-lightmicro- the most significant difference between the two types of
scope. spectra: itisapproximately0.8or0.5,respectively,when
thephotonpolarizationsareperpendiculartoorpartially
along the hexagonal c-axis. To demonstrate how R can
B. Identification of individual domain orientation be used to characterize surface-domain structures, we
have performed space-resolved Raman scattering mea-
Optical images of crystal faces can only provide infor- surements on each of the four types of crystal faces that
mation about the relative orientations of the hexagonal are labeled as A, B, C, and D in Figs. 1(e-f). The data
c-axisinadjacentdomains(Figs.1(a-b)). Othermethods aredisplayedinFigs.3(a2-d2),alongwithpolarized-light
arerequiredtorevealthe absolute orientationofindivid- optical images of the surfaces in the top panels (a1-d1).
ual domains. We reported in our previous work19 that Intheopticalimagesahorizontalwhitelineindicatesthe
RamanscatteringcandetectA andE opticalphonons trajectory on which the Raman spectra were taken. In
g g
separately with parallel and perpendicular combinations these Raman measurements, the incident and scattered
of incoming- and scattered-photon polarizations, respec- photonpolarizationsarealwayskepttobeparallel,along
tively,whenatleastoneofthepolarizationsisparallelto one of the two perpendicular face diagonals ([110]c and
thehexagonalc-axis. Ifnoneofthepolarizationsisalong [110]c, labeled as “XX” and “YY”, respectively) on the
the hexagonal c-axis, the acquired Raman spectrum will surface.
be a weighted combination of A and E signals.18 The Indeed, on A-type (Figs. 3(a1-a2)) and D-type
g g
relative intensities of A and E phonon peaks depend (Figs. 3(d1-d2)) faces, R is found to switch between 0.5
g g
on the exact scattering geometry. Figure 2 displays Ra- and0.8everytimethescanningpositioncrossesabound-
manspectratakenunder“XX” and“YY”geometrieson ary between the stripes, and whenever R is around
XX
4
Forcrystalswith{100} domainwalls(Figs.1(a,e)),the
c
strain tensors of the two domains are S and S (Eq. 2);
1 2
1 when they are contractedwith the vector (1,0,0),which
) EgAg Ag Eg u.)0.8 lies within all A-type surfaces and crosses the domains,
units I (a. 0.5 rthesepeocuttivcoelmy)eshaovfeSo1papnodsiteS2pr(o(0je,cdt,iodn)saanldon(g0,b−odth,−tdh)e,
b. 350 400 450 [010]c and [001]c directions. These are the normal direc-
ar ω(cm-1) tions of the A-type surfaces, and thus the surfaces are
y ( wrinkled. A similar argument can be used to explain
sit XX thewrinklesonC-typesurfaces(Figs.1(b,f)), wherethe
n straintensorscanbetakenasS andS ,whicharetobe
e 2 3
nt contracted with the vector (1,−1,0) and then projected
I
along [001] . For D-type surfaces, we should instead use
c
YY (1,0,0) (or (0,1,0)) as the vector to be contracted with
S and S , but here the outcomes have the same pro-
2 3
100 200 300 400 500 600 700 800 900 jection along the [010] (or [100] ) direction, hence the
c c
-1 D-type surfaces are not wrinkled despite the presence
Raman shiftω(cm )
of domain walls. No wrinkles are expected on single-
domain B-type surfaces. Using the room-temperature
lattice constants of CaMn O ,32 the angles of wrinkles
7 12
FIG. 2. (Color online) Polarized Raman spectra obtained at ◦
onA-andC-typesurfacesarecalculatedtobe 0.75 and
room temperature, offset for clarity. Data curves are color- ◦
1.05 ,respectively,whichareingoodagreementwithour
coded with arrows in the upper-left inset that indicate the
polarization geometries with respect to the hexagonal c-axis surface-profile data.
(magenta body diagonal of the cube). Vertical dashed lines Taking the above results altogether, we suggest a
indicate Ag and Eg phonon peaks.19 Upper-right inset, XX new method to determine ferroelastic domain structures
andYYspectrainthe350to450cm−1 region,normalizedat basedonmeasurementsof only one pseudo-cubic sample
425 cm−1. surface. The most reliable way is to use Raman spec-
troscopy combined with surface profile measurements,
with polarized-light microscopy being a complementary
0.5, R is around 0.8 (and vise versa). The results but not necessary method. First, one needs to find the
YY
are in perfect agreementwith our expectations based on orientation of the domain walls by scanning in different
the model illustrated in Fig. 1. On the other hand, R directions and rotating photon polarizations in the Ra-
is found to remain roughly constant in a given scatter- man scattering measurement, aiming to maximize the
ing geometry across the entire B-type (Figs. 3(b1-b2)) contrast in both the Raman and surface profile data.
and C-type (Figs. 3(c1-c2)) faces, for different reasons: Second, the surface domain structure can be know by
theB-typesurfacesaresingle-domain,whereasontheC- comparing the data with the results shown in Fig. 3.
type surfaces the hexagonal c-axes in different domains Third, from the surface profile data one can further tell
have the same projection onto the surface plane. These apart the aforementioned [111]c and [111]c degenerated
resultsrenderRamanspectroscopy,whenusedalone,un- situations onA- andC-type surfaces,using the fact that
able to distinguish between A-type and D-type, and be- shortened hexagonal c-axis always connects the valleys
tween B-type and C-type surfaces. In combination with of the wrinkles. Our method is particularly useful when
polarized-lightmicroscopyonecaneasilytellthemapart, the sample is in thin-film form, or when the edges of
butneitheropticalinspectionnorRamanspectroscopyis crystals are not along a high-symmetry direction. While
able to distinguish, e.g., between hexagonal c-axis orien- piezoresponseforcemicroscopy33 ismostcommonlyused
tations of [111] and [111] , where the degeneracy is due tostudyferroelectricdomainstructures,ourmethodpro-
c c
to the fact that light is propagating along the [001] di- videsaroutetomonitoringtheferroelasticdomainsboth
c
rection. aboveandbelowtheferroelectrictransitiontemperature,
which may help improve our understanding of the inter-
A feasible way to completely determine the domain
play between different ferroic order parameters in multi-
structure by looking at only one surface is to measure
ferroics. Todetectnano-scaledomains,tip-enhancedRa-
alternatinginclinations,orwrinkles,onthe surface. Fig-
manspectroscopy34,35 canbeusedtoenhancethespatial
ures 3(a3-d3) display surface profiles measured roughly
resolution of our method.
along the same trajectory on which we took the Raman
spectra. WefindthattheA-typeandC-typesurfacesex-
hibitclearzigzagprofiles,whereastheB-typeandD-type
C. Domain switching and detwinning effect
surfaces are essentially flat. Moreover, the angles of the
◦
zigzag profiles on the A-type (≈ 0.80(6) ) and C-type
◦
(≈ 1.05(6) ) surfaces are slightly different. To under- Inadditiontomethodsforcharacterizingthedomains,
stand these results, we refer to the schematics in Fig. 1. wefoundthatdomainstructureinCaMn O singlecrys-
7 12
5
Y X Y X Y Y X
X
(a1) (b1) (c1) (d1)
y
t
ensi 0.8 XX YY YY YY
t
n
e I
v 0.6
ti
a
Rel 0.4 YY (a2) XX (b2) XX (c2) XX(d2)
2
)
m
n 1
20
1
ht ( 0 179.20° 178.95°
g
ei -1
H (a3) (b3) (c3) (d3)
-2
0 50 100 0 50 100 0 50 100 0 50 100
Position µ(m) Position µ(m) Position µ(m) Position µ(m)
FIG. 3. (Color online) (a1)-(d1) Optical images of A-, B-, C-, and D-type crystal faces (Figs. 1(e-f)). Horizontal white line
indicates the trajectory along which the measurements in (a2)-(d2) and (a3)-(d3) are performed. Half-solid-half-dashed line
indicatestheorientationofthehexagonalc-axis(alongoneoftheh111ic directions)ineachdomain,withthesolidendpointing
atthetopface. (a2)-(d2)RamanintensityratioR(seetext)measuredalongthetrajectoriesindicatedin(a1)-(d1),respectively,
with different photon polarizations. The data are color-coded with the arrows indicating polarization directions in (a1)-(d1).
(a3)-(d3) Surface profiles along thetrajectories indicated in (a1)-(d1).
tals canbe alteredatroomtemperatureunder moderate sonably expect that a h111i domain can be exclusively
c
uniaxial compression. Figure 4 displays the same face of selected if the compression is primarily along a body di-
a crystalatdifferenttimes. Initially,the surfaceexhibits agonal of the cube, as is indeed shown to be the case in
a stripe pattern indicative of presence of {100} domain Fig.4(d). Apartfromtakingopticalimages,wehaveper-
c
walls (Fig. 4(a)). When a compressiveforce of about 1.5 formed Raman scattering measurements on all six faces
N is applied along the direction shown by the arrows in of the sample in Fig. 4(d), which consistently show that
◦
Fig.4(b),thestripepatternisrotatedby45 ,whichindi- the crystal is highly detwinned.
cates the formation of new {110}c domain walls and the This switching behavior of ferroelastic domains in
disappearanceofthe oldones. The magnitude ofthe ap- CaMn O stems from the fact that the rhombohedral
7 12
pliedforce amountsto a uniaxialstressofabout30 MPa distortion is characterized by the shortening rather the
inside the crystal, and the new stripe pattern persists elongation of a body diagonal, which makes the detwin-
after the force was removed. Similarly, when a compres- ning operations simple. Among other rhombohedrally
sionis appliedalongthe directionin Fig.4(c), the stripe distortedperovskites,LaAlO 36 is similartoCaMn O ,
3 7 12
◦
pattern is found to rotate again by 90 . This domain- whereas the distortion in BiFeO features an elongated
3
switching behavior can be understood as the following: bodydiagonal,makingitrathertrickytopreparecertain
when a uniaxial compression is applied, the ferroelas- types of domain structure.37,38
tic domainsrearrangethemselves to minimize the length
along the direction of compression. In the case of com-
pression in the [110] direction, domains with hexagonal
c IV. CONCLUSIONS
c-axes along [111] and [111] are energetically favored.
c c
According to the analysis in Section IIIA, this may re-
Insummary,wehaveobservedandinvestigatedferroe-
sult in the formation of either (001) or (110) domain
c c
lastic domain structures in single crystals of CaMn O .
walls,buttheformerwouldalsogeneratewrinklesonthe 7 12
For cube-shaped single crystals with multiple domains,
(100) and (010) faces which are incompatible with the
c c
we can determine the orientation of individual domains
applied compression. Hence the resulting domain walls
by measurements either on two adjacent faces with Ra-
are parallel to (110) . For the same reason, one can rea-
c
man spectroscopy alone, or on only one face using both
6
Ramanspectroscopyandstylussurfaceprofiler. Thelat-
(a) (b) termethodisalsosuitablefordeterminingdomainstruc-
turesinthin-filmsamples,whereonlyonesurfaceisavail-
able. In addition, polarized-light microscopy provides a
complementary and convenient way to observe the do-
main structure. Finally, we find that the domain struc-
ture can be alteredby moderate uniaxialcompressionat
room temperature, which allows for a simple method to
obtain twin-free samples with a controlled orientationof
thehexagonalc-axis. Ourresultsoffertheopportunityto
(c) (d) prepare well-defined CaMn O samples, e.g., for spec-
7 12
troscopicstudiesthatrequiresingle-domaincrystals,and
the methods we use can be readily transferred to stud-
ies of thin-film samples as well as other rhombohedrally
distorted cubic compounds.
Acknowledgments
FIG. 4. (Color online) Optical images of the same face of
a crystal: (a) initial state, (b) after compression was applied We thank W. H. Yang and X. B. Wang for technical
alongthefacediagonalindicatedbythearrows,(c)aftercom- assistance in the use of polarized-light microscope and
pression was applied along the other face diagonal, and (d) stylus profiler. This work is supported by the NSF of
detwinned state after compressed primarily along a body di- China (No. 11374024) and the NBRP of China (No.
agonal. 2013CB921903).
∗ [email protected] (2012).
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