Experimental evidence for field induced emergent clock anisotropies in the XY pyrochlore Er$_2$Ti$_2$O$_7$

3.7 MB·

by  J. Gaudet

Experimental evidence for field induced emergent clock anisotropies
in the XY pyrochlore Er Ti O
2 2 7
J. Gaudet,1,∗ A. M. Hallas,1 J. Thibault,1 N. P. Butch,2 H. A. Dabkowska,3 and B. D. Gaulin1,3,4
1Department of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada
2NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA
3Brockhouse Institute for Materials Research, Hamilton, ON L8S 4M1 Canada
4Canadian Institute for Materials Research, 180 Dundas Street West, Toronto, Ontario M5G 1Z8, Canada
(Dated: January 13, 2017)
TheXYpyrochloreantiferromagnetEr Ti O exhibitsararecaseofZ discretesymmetrybreak-
2 2 7 6
inginitsψ magneticgroundstate. Despitebeingwell-studiedtheoretically,systemswithhighdis-
2
crete symmetry breakings are uncommon in nature and, thus, Er Ti O provides an experimental
7 2 2 7
playground for the study of broken Z symmetry, for n > 2. A recent theoretical work examined
1 n
the effect of a magnetic field on a pyrochlore lattice with broken Z symmetry and applied it to
0 6
Er Ti O . Thisstudypredictedmultipledomaintransitionsdependingonthecrystallographicori-
2 2 2 7
entation of the magnetic field, inducing rich and controllable magnetothermodynamic behavior. In

n this work, we present neutron scattering measurements on Er Ti O with a magnetic field applied
2 2 7
a alongthe[001]and[111]directions,andprovidethefirstexperimentalobservationoftheseexoticdo-
J
maintransitions. Ina[001]field,weobserveaψ toψ transitionatacriticalfieldof0.18±0.05T.
2 3
1 Wearethusabletoextendtheconceptofthespin-floptransition,whichhaslongbeenobservedin
1 Isingsystems,tohigherdiscreteZ symmetries. Ina[111]field,weobserveaseriesofdomain-based
n
phase transitions for fields of 0.15 ± 0.03 T and 0.40 ± 0.03 T. We show that these field-induced
]
transitions are consistent with the emergence of two-fold, three-fold and possibly six-fold Zeeman
l
e terms. Consideringallthepossibleψ andψ domains,theseZeemantermscanbemappedontoan
2 3
-
analog clock - exemplifying a literal clock anisotropy. Lastly, our quantitative analysis of the [001]
r
t domain transition in Er Ti O is consistent with order-by-disorder as the dominant ground state
s 2 2 7
. selection mechanism.
t
a
m PACSnumbers: 75.25.-j,75.10.Kt,75.40.Gb,71.70.Ch
-
d
1. INTRODUCTION nario is not yet definitive, however, as a competing the-
n
ory has been proposed that might explain the ground-
o
c state selection via energetic selection [10–12]. Regard-
[ The pyrochlore lattice is a face centered cubic struc- less,Er Ti O remainsthemostpromisingcandidatefor
2 2 7
ture with a basis of corner-sharing tetrahedra. In the ground state selection via the order-by-disorder mecha-
1
v case of the pyrochlore Er2Ti2O7, the spins residing on nism.
5 this network of tetrahedra are known to have a k = 0,
2 Γ magnetic structure, for which all the spins lie in the Another interesting aspect of the phase transition in
5
1
plane perpendicular to the local <111> axis [1]. A rep- Er Ti O is that it represents a rare case of higher or-
3 2 2 7
resentationofthisΓ structureisshowninFig.1(a)with der discrete symmetry breaking. Indeed, time inversion
0 5
1. its associated basis vectors ψ2 and ψ3. The linear com- and rotational symmetry allow six distinct spin orienta-
binationofthesetwobasisvectorscangenerateanyspin tions within the ψ state, ie. a Z symmetry breaking.
0 2 6
7 orientation spanning the local XY plane, which is the The Ising model, with Z2 symmetry breaking, is known
1 entire U(1) manifold. An appropriate model Hamilto- to capture the salient physics of many magnetic materi-
: nian that includes anisotropic exchange and dipolar in- als and a wealth of other physical systems [13, 14]. Al-
v
i teractions,withexperimentallydeterminedexchangepa- though many theoretical models for systems with higher
X
rameters for Er Ti O , has its energy minimized by the Z symmetry breaking have been proposed, there are
2 2 7 n
r U(1) manifold, which is degenerate at the mean field in fact few experimental realizations of higher discrete
a
level [2, 3]. However, in the real material, this degen- symmetry [15, 16] Thus, Er Ti O provides a rare and
2 2 7
eracy is lifted when the Er3+ moments order antiferro- possiblyuniqueopportunitytoinvestigatetheproperties
magnetically into a pure ψ state below T =1.2 K [4]. of such discrete symmetry breaking. Recently, Maryasin
2 N
The mechanism responsible for this degeneracy breaking et al. [17] developed a theor,y for the e,ffect of a mag-
in Er Ti O has attracted much attention, as it could netic field on the properties of a Z pyrochlore magnet
2 2 7 6
be the first demonstration of ground state selection via and predicted rich and exotic domain effects to occur in
order-by-disorder[5,6]. Indeed,itiswidelybelievedthat Er Ti O due to the emergence of two-fold, three-fold
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instead of ordering via energetic selection, thermal and and six-fold anisotropic Zeeman terms. Th,e degeneracy
quantumfluctuationsdrivethissystemtoanentropically as well as the local XY angle of the states minimized by
favorablemagneticallyorderedstate[1–3,7–9]. Thissce- each of these Zeeman terms can be mapped out in anal-
2
(a) 𝛙2 1112 1 (b) <111> and the (K+H,K−H,−2K) scattering planes using x-ray
10 2 𝛙2 Laue diffraction. Time-of-flight neutron scattering mea-
9 3
8 4 surements were performed using the Disc Chopper Spec-
7 6 5
trometer (DCS) at the NIST Center for Neutron Re-
search[21]. Anincidentwavelengthof5˚Awasemployed,
𝛙3
givingamaximumenergytransferof∼2meVandanen-
ergyresolutionof0.09meV.Alluncertaintiescorrespond
(c) <111>
to one standard deviation. A magnetic field was applied
𝛙3 perpendiculartothescatteringplane. Thus,forthesam-
a
ple aligned in the (H,K,0) scattering plane, the field is
applied along the [001] direction. For the sample aligned
b inthe(K+H,K−H,−2K)scatteringplane,thefieldisap-
plied along the [111] direction. For both alignments, the
FIG. 1: (a) The k = 0, Γ magnetic structure on the py- (220) or (2-20) Bragg peak was observed only within the
5
rochlore(Fd¯3m)lattice,whichisconstructedbytheψ2 (red) central bank of our detectors providing a 2◦ upper limit
and ψ3 (blue) basis vectors. Within the local XY plane, six onthepossiblemisalignmentofourmagneticfield. Scans
specific spin orientations are allowed by (b) ψ2 and six inter- with a total sample rotation of 35° with 0.25° steps were
leaving angles are allowed by (c) ψ . The ensemble of the
3 performed, centered on the (220) or (2-20) Bragg peak
full ψ and ψ states mimic a literal clock and can be used
2 3
(these positions are symmetrically equivalent and will
to represent the different anisotropic Zeeman terms via the
selection of different hours (angles) on the clock. henceforth be referred to as (220)). Lastly, for complete-
ness, wealsopresentpreviouslypublishedmeasurements
ofEr Ti O alignedinthe(H,H,L)scatteringplane,with
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ogy to a conventional clock, where the twelve hours of a magnetic field applied along the [1-10] direction. This
the clock are represented by the six ψ and the six ψ earlierexperimentwasperformedonthesamespectrom-
2 3
states (Fig. 1(b,c)). eter (DCS at NIST) but with a different single crystal;
In this paper, we use time-of-flight neutron scattering the full experimental details can be found in Ref. [22].
on Er Ti O with a magnetic field applied along high
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symmetry cubic directions to provide the first experi- TypicalelasticscatteringmapsofEr Ti O foreachof
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mental evidence of these predicted domain effects. We thethreesampleorientationsabove(T =8Kor2K)and
find a host of low field domain selections and reorien- below (T = 60 mK or 30 mK) the Neel ordering transi-
tations, which henceforth we will collectively refer to as tionareshowninFig.2. Athightemperature,weobserve
“domain transitions”, not to be confused with a change a resolution limited Bragg peak that is purely structural
of representation manifold (U(1) or Γ5) which would be in origin. At low temperature, passing into the mag-
a thermodynamic phase transition. Indeed, we find that netically ordered state, additional magnetic Bragg and
forafieldappliedalongthe[001]direction,Er2Ti2O7 ex- diffuse scattering can be observed at the (220) position
hibits a clear ψ2 to ψ3 transition at a critical field of for all sample orientations. In the subsequent analysis,
0.18±0.05T.Thisdomaintransitioncanbeseenasthe elastic cuts have been extracted for each data set with
Zn generalization of the spin-flop transition that occurs varying magnetic field, as indicated by the white dashed
in Ising Z2 systems. Our neutron scattering results also lines in Fig. 2(a,c,e). Those cuts have been obtained by
indicate possible domain transitions at 0.15 ± 0.03 T integrating the respective data sets in energy from −0.1
and 0.40 ± 0.03 T in a [111] magnetic field. We pro- to 0.1 meV with an additional integration (i) from −0.3
vide a complete description of the domains transitions to 0.3 in the [0,0,L] direction for the [1-10] sample, (ii)
thatoccursinEr2Ti2O7 insuchfieldsandshowthatour from1.8to2.2inthe[H00]directionforthe[001]sample,
observations are consistent with the predicted emergent and(iii)from0.8to1.2inthe[H’,0,-H’]directionforthe
Zeeman two-fold, three-fold and possibly six-fold clock [111] sample (Fig. 3(a,c,e)). The inelastic spectra, which
terms. are shown for each field direction in Fig. 4, are extracted
by using the same directional integrations, but without
the integration in energy. Integrations of the total in-
II. EXPERIMENTAL DETAILS elastic signal about (220) are presented in Fig. 5, where
the area of integration corresponds to the white dashed
A large single crystal of Er Ti O was grown in a boxes in Fig. 2(b,d,f). Those spectra are obtained using
2 2 7
floating zone image furnace in 3 atm. of air and with the same directional integrations as above, but with an
a growth rate of 7 mm h−1. This method of crystal additional integration (i) from 1.7 to 2.3 in [H,H,0] for
growth is well-established for the rare earth titanate py- the [1-10] sample, (ii) from 1.8 to 2.2 in [0K0] for the
rochlores[18–20]. Thiscrystalwascutintotwo2-3gram [001] sample, and (iii) from 0.8 to 1.2 in [K’,-2K’,K’] for
segments, whichwererespectivelyalignedinthe(H,K,0) the [111] sample.
3
III. RESULTS 1 10 100
A. Magnetic Field Dependence of the Elastic (a)T = 2K (b)T = 30mK
2
Scattering at (220) [H,H,L] Plane [H,H,L] Plane
L] 1
Neutron scattering spectra of Er2Ti2O7 were collected 0,0,
[
at very low temperature, below 100 mK, with a mag- 0
netic field ranging from 0 T to 3 T applied along three
crystallographic directions: [1-10], [001], and [111]. As -1
0 1 2 0 1 2
can be seen by comparing the high temperature data [H,H,0] [H,H,0]
sets (Fig. 2(a,c,e)) with the low temperature data sets
(c) (d)
(Fig.2(b,d,f),thereisconsiderablemagneticdiffusescat-
3
T = 8K T = 60mK
tering at low temperature for all field directions. This
diffuse scattering is far broader than a resolution lim- K,0] 2 [H,K,0] Plane [H,K,0] Plane
ited Bragg peak typical of long range order. The origin 0,
[
of this diffuse scattering is an intense quasi Goldstone- 1
mode, which softens towards (220) [2, 11, 22]. The
0
quasi Goldstone-mode excitations have been previously 0 1 2 0 1 2
[H,0,0] [H,0,0]
measured in detail and are known to be gapped by
1
0.053 ±0.006 meV [11, 23]. As the energy resolution
(e)T = 8K (f) T = 60mK
of this experiment is 0.09 meV, we inevitably integrate [K+H,K-H,-2K] 
 [K+H,K-H,-2K] 

overaportionofthislowenergyinelasticscatteringwhen K] Plane Plane
2
extracting the elastic component. Thus, the diffuse scat- K,-
tering observed in Fig. 2(b,d,f) originates from a partial K,
[ 0
integration of the quasi Goldstone-mode magnetic exci-
tations.
0 1 2 0 1 2
To investigate the field dependence of the elastic scat-
[H,-H,0] [H,-H,0]
tering, we performed integrations along the (220) Bragg
peak for each data set at low temperature, with varying FIG. 2: The elastic scattering of Er Ti O above and be-
2 2 7
field strength and direction. A representative selection low the Neel ordering transitions for crystals aligned in
oftheseelasticscatteringcutsareshowninFig.3(a,c,e). the (a,b) (H,H,L) plane, (c,d) (H,K,0) plane, and (e,f)
At low temperature, there is a significant increase in the (K+H,K−H,−2K) plane. At high temperature, the inten-
sityofthe(220)Braggpeakispurelystructural,whileatlow
intensity of the resolution-limited Bragg scattering, due
temperature a magnetic Bragg peak also forms on the (220)
tothelongrangemagneticorder,andalso,adiffusecon-
position. Eachofthesedatasetshasbeenintegratedinenergy
tribution originating from the inelastic scattering as dis-
from−0.1to0.1meV,whichisapproximatelytheresolution
cussed above. These two contributions to the scattering of elastic scattering in this experiment. Within the magneti-
at the magnetic Bragg position necessitate a two com- cally ordered state, this integration picks up a component of
ponent fit [24]. These fits were performed with a Gaus- the magnetic inelastic scattering, giving the appearance of a
sianfunction,toaccountfortheresolutionlimitedelastic significantlybroadenedpeak. Thedashedwhitelinesindicate
the areas of integration described in the text.
Bragg scattering, and a Lorentzian function, which cap-
turestheinelasticcontributiontothescattering. TheQ-
widthoftheelasticGaussianpeakwasfixedtothevalue
determined from fitting the high temperature data set, netic Bragg peak in a [1-10] field is shown in Fig. 3(b).
forwhichthescatteringat(220)ispurelystructural. The At low field, we observe an abrupt doubling of the in-
width of the Lorentzian function was allowed to freely tensity at (220). Above 0.5 T, the elastic scattering
vary. A sloping background was also used to account smoothly diminishes as a function of field and reaches
for the instrumental background. The solid lines seen zero intensity by 1.5 T. This smooth diminution of the
inFig.3(a,c,e)areexamplesoftypicalfitsobtainedfrom (220) elastic scattering at high field corresponds to the
followingthisprocedure,givingexcellentagreementwith transition towards the field polarized state [17, 22, 25–
themeasureddata. Theresultingintegratedintensityfor 27]. For the [001] field direction, the magnetic elastic
theelastic(Gaussian)componentasafunctionofthe[1- intensity at (220) falls off precipitously under the appli-
10], [001], and [111] applied magnetic field are shown in cation of a small field (Fig. 3(d)). Indeed, the elastic
Fig. 3(b,d,f), respectively. The field dependence of the magnetic scattering at (220) reaches zero intensity in a
Lorentzian part of the scattering is presented in the Ap- field as small as 0.2 T. As the field is further increased,
pendix A for all field directions. up to 3 T, the intensity remains zero. Finally, for the
The elastic scattering dependence of the (220) mag- [111]field,theelasticscatteringat(220)isunaffectedup
4
and [111] field. Integrations as a function of energy cen-
tered on the (220) Bragg peak for each field direction
are also shown in Fig. 5. The spin wave spectra at 0 T
is dominated by low energy quasi Goldstone-mode exci-
tations centered on (220). These low energy excitations
have a linear dispersion and are maximally intense ap-
proaching the (220) Bragg position. Accordingly, in the
0 T energy cuts of Fig. 5, the integration over the quasi
Goldstone-mode excitations produces the first inelastic
feature, centered just below 0.2 meV. At slightly higher

energy,0.35meV,wealsoobserveweakerflatmodes. The
structure of the spin wave spectra in an applied field are
furtheranalyzedin,first,thelowfieldregimeandsecond,
in the high field regime crossing into the field polarized
state. Thecriticalfieldofthetransitiontothefieldpolar-
ized state depends on the field orientation and is known,
from bulk measurements, to be 1.5−1.7 T [25, 26].
Theapplicationofasmall[1-10]fieldresultsinacom-
pleteabsenceoflowenergyscatteringat0.5T,indicating
the removal of the quasi Goldstone-mode excitations at
(220) (Fig. 4(a) and 5(a)). However, the opposite sce-
nario is observed upon the application of a weak [001]
field. Indeed, an immediate increase in the scattering of
the quasi Goldstone-mode excitations is observed. This
FIG.3: Representativeselectionofelasticcutsoverthe(220) enhancement of the scattering can be seen by comparing
Bragg peak in varying magnetic field strength, for fields ap- the 0 T and 0.5 T data sets in Fig. 4(b) and 5(b). For
plied along (a) the [1-10] direction, (c) the [001] direction, the [111] field direction, the quasi Goldstone-mode exci-
and (e) the [111] direction. The solid lines in these panels tationsshowsnofielddependenceupto0.15T(Fig.4(c)
are the fits to the Bragg peak, which were used to extract
and 5(c)). Above 0.15 T, we observe a suppression of
the integrated intensity. The resultant field dependence of
thequasiGoldstone-modesexcitations,butwithouttheir
the magnetic elastic intensity at (220) with the field applied
complete removal, as in the case of a [1-10] field. In fact,
alongthe(b)[1-10],(d)[001],and(f)[111]direction,revealing
multiple low field domain transitions in Er Ti O . The red while continuously decreasing up to 1 T, the intensity of
2 2 7
curves in these panels corresponds to the theoretically pre- these excitations remains finite in a [111] field.
dicted domain transitions [17]. Note that the (2-20) Bragg
positionissymmetricallyequivalentto(220),asitisreferred
to in the main text. Lastly, we can examine the changes in the spin wave
spectra upon transitioning into the field polarized state,
whichisknowntooccurat1.5−1.7T[25,26]. Theevo-
to 0.15 T (Fig. 3(f)). Between 0.15 T and 0.4 T, the lutionofthespinwavespectrawhenpassingintothefield
elastic scattering abruptly increases, reaching an inten- polarized state gives similar behavior for [1-10] and [111]
sity that is 1.75 times larger than the elastic intensity fields. Fig.4(a)and(c)showthatthespectralweightfor
at 0 T. The intensity then remains constant from 0.4 T these two field orientations softens towards the elastic
to 1 T. Above 1 T, the intensity decreases, ultimately line in a 1.5 T field. Upon further increasing the mag-
reaching zero intensity for fields larger than 1.5 T. This netic field to 3 T, this quasi-elastic scattering moves to
decrease of the elastic scattering at (220) above 1.5 T is, higher energies, and forms weakly-dispersing spin wave
asbefore,concomitantwiththephasetransitiontowards modes. A qualitatively different behavior is observed
the field polarized state [25, 26]. in the case of a [001] field. Approaching the polarized
statefora[001]field,coherentlowenergyexcitationsare
still observed, but with an increasing spin wave gap at
B. Magnetic Field Dependence of the Inelastic (220). In Fig. 5(b), the opening of the spin wave gap
Scattering at (220) is demonstrated by the shifting of the low energy fea-
ture upwards in energy. Upon the application of a [001]
Now, turning our attention to the inelastic scatter- field greater than 1.5 T, the dispersion of the low energy
ing, we can first look at Fig. 4, which shows the spin quasi Goldstone-mode excitations smoothly evolve to a
wave spectra of Er Ti O as a function of [1-10], [001], non-dispersive mode, as seen at 3 T.
2 2 7
5
V) 0.8(a) H || [1-10] 0 T 0.5 T 1T 1.5T 2T 3T 1
e
m
y (
g
er
n 0
E
1.5 2 2.51.5 2 2.51.5 2 2.51.5 2 2.51.5 2 2.51.5 2 2.5
[H,H,0]
meV) 0.6(b) H || [001] 0 T 0.5 T 1T 1.5T 2T 3T Inte
gy ( nsity
er (a
n 0 .u
E .)
1 2 3 1 2 3 1 2 3 1 2 3 1 2 31 2 3
[2,K,0]
V) 0.6(c) H || [111] 0 T 0.5 T 1T 1.5T 2T 3T
e
m
y (
g
er
n 0
E
0 1 2 0 1 2 0 1 20 1 2 0 1 20 1 2 0
[1+K’,-2K’,K’-1]
FIG.4: MagneticfielddependenceoftheenergyslicesofEr Ti O centeredonthe(220)Braggposition. Theseenergyslicesare
2 2 7
plottedalong(a)the[H,H,0]directionfora[1-10]field,(b)the[0,K,0]directionfora[001]field,and(c)the[K’,-2K’,K’]direction
for a [111] field. For the [1-10] field orientation, there is an immediate diminution of the quasi Goldstone-mode excitations at
0.5T.Forthe[001]fieldorientation,thequasi-Goldstonemodeexcitationsat(220)areintensifiedbytheapplicationofafield
up to 1 T. In a [111] field, the quasi-Goldstone mode excitations have their intensity continuously decreased for fields ranging
from 0.15 to 1 T. Above 1.5 T, the response of the spin wave spectra is due to the transition towards the field polarized state.
IV. DISCUSSION Thus,byusingtheresultsofthecalculationinFig.6and
thefactthatthemagneticstateat0Tiswellknown(six
ψ states), it is possible to deduce the reorientation of
2
The magnetic ground state of Er Ti O in zero mag- the domains that occur in a magnetic field by measuring
2 2 7
netic field is well-established and corresponds to an the relative change of the (220) elastic intensity.
equiprobable distribution of the six ψ domains within
2
Γ [4]. Theelasticneutronscatteringprofileofthismag-
5
netic structure is characterized, in part, by an intense A. Domain Selection in a [1-10] Magnetic Field
(220)magneticBraggpeak[1,10,22]. Onceaweakmag-
netic field is applied along any direction, the degeneracy Before discussing the [001] and [111] field evolution
of the six ψ domains is lifted. However, it is impor- of the (220) Bragg peak, we briefly review the well-
2
tant to re-emphasize that this does not correspond to a established domain effects for a field applied along the
thermodynamicphasetransitionorachangeofrepresen- [1-10] direction. For this field orientation, an increase
tation manifold. Rather, the spins remain constrained of the scattering at (220) is observed for fields above
to the U(1) plane (Γ manifold), as has been previously 0.1 T (Fig. 3(b)) [1, 22]. The origin of this intensity
5
shown by heat capacity [22, 26], magnetization [25], and gain is well-understood: the application of a [1-10] field
neutron scattering [22, 27]. Depending on the field ori- on Er Ti O in its ψ magnetic ground state induces a
2 2 7 2
entation, this degeneracy breaking results in an increase two-foldclocktermthatfavorstheψ stateswithXYan-
2
or decrease of the (220) Bragg peak intensity due to do- gles of 0 and π [2, 17]. These two angles are highlighted
main effects. To understand these intensity changes, it by the dashed black circles in Fig. 6(a), and are the ones
is important to understand that the scattered intensity that maximize the scattered intensity of the (220) mag-
at (220) follows an I ∝ cos2(θ) relationship, where θ netic Bragg peak. These two angles give a factor two
is the local XY angle (Fig. 6). As neutron scattering intensity increase to (220) from theaverage value for the
is only sensitive to the component of the magnetization six zero field ψ states. These two domains selected by
2
perpendicular to the direction of the scattering vector, the Zeeman-clock term for a [1-10] field are also selected
a variation in the scattered intensity is observed due to by the six-fold anisotropic term at 0 T. No further do-
thedifferentorientationsofthemomentsineachdomain. main transitions are observed or predicted at low field.
6
FIG.5: Theinelasticintensitycenteredon(220)asafunctionofenergyforvarying(a)[1-10]magneticfields,(b)[001]magnetic
fields and (c) [111] magnetic fields. In a [1-10] field, the low energy scattering is completely suppressed upon the application
of a 0.5 T field. In a [001] field, the inelastic intensity at low energies strongly increases for fields as small as 0.1 T. Further
increasing the field shifts the spectral weight to higher energies. In a [111] field, the inelastic spectra is unaffected by fields up
to0.15T.Atlarger[111]fields,theintensityatlowenergiescontinuouslydecreases. TheintegrationperformedinQaregiven
in the experimental methods section.
Beyond that, at high field, a continuous transition to- tion between the emergent two-fold Zeeman term and
wardsthefieldpolarizedstateisobservedat1.5Twhich the 0 T six-fold clock term. At higher fields, as the Zee-
isindicatedbythesmoothdiminutionofthe(220)inten- man energy begins to overwhelm the 0 T six-fold clock
sity (Fig. 3(b)). This diminution of the (220) intensity term, there will be a continuous rotation of the spins
occurs due to canting effects that become non-negligible towards a pure ψ state made up of π/2 and 3π/2 do-
3
approaching the field polarized state. This canting ef- mains, wherethe(220)intensityshoulddecreasetozero.
fectintroducesanon-zerospincomponentawayfromthe In our experiment, the intensity of (220) continuously
XY local plane. Several studies have modeled this be- anddrasticallydecreasestoreachzerointensityby0.2T
havior [10, 22, 27], but we reproduced via the red line in (Fig. 3(b)). We associate this dramatic intensity loss
Fig.3(b),theresultsusingthemethodfoundinMaryasin with the above-described ψ to ψ transition, which is
2 3
et al. [17] which capture well the experimental data. an XY spin-flop transition. As opposed to the ψ state,
2
all the spins within the ψ states point perpendicular to
3
the [001] direction (see Fig. 1(a)). This result general-
B. Domain Selection and Reorientation in a [001] izes the well known concept of spin-flop transitions seen
Magnetic Field in Ising system (Z ) to systems with discrete symmetry
2
breaking Z with n>2.
n
Aswasthecasefora[1-10]field, ithasbeenpredicted The domain transition in a [001] field also opens an
thatina[001]field,theZeemancouplingwillgiveriseto interesting line of inquiry, as the critical field for the ψ
2
a two-fold clock term [17]. The states that are selected to ψ transition occurs when the Zeeman energy equals
3
in a [001] field are rotated by π/2 with respect to the the energy of the 0 T six-fold clock term. Thus, we are
statesfavoredbya[1-10]field. Thus,theZeemanenergy able to provide an independent measurement of the 0 T
is fully minimized by the ψ states with XY local angles six-fold clock term in Er Ti O , which is of course rele-
3 2 2 7
of π/2 and 3π/2. However, at very low fields, where the vant to the zero field ground state selection, be it order-
Zeemanenergyismuchsmallerthanthesix-foldψ clock by-disorder and/or energetic selection involving virtual
2
term that dominates at 0 T, it is not possible to select crystal field processes. Using analytical equations from
these two ψ domains. Instead, the system compromises Ref. [17], which account for the field dependence of the
3
byselectingthenearestψ domains,thosewithXYlocal XY local angle upon the application of a [001] field, it is
2
angles of 2π/6, 4π/6, 8π/6 and 10π/6 (See black dashed possible to model the decrease of the elastic scattering,
circles on Fig. 6(b)). Compared to 0 T, this new domain the result of which is shown by the red line in Fig. 3(d).
distribution should decrease the (220) intensity by a fac- Thisfitisoptimizedasafunctionofthecriticalfield,and
tor two. Experimentally, referring back to Fig. 3(b), we thevalueweobtainedis0.18±0.05T.Calculationsusing
do indeed observe a clear decrease of the scattering at only the quantum order-by-disorder term have predicted
low field. a critical field of 0.2 T [17], which is in excellent agree-
Theapplicationofafieldalong[001]isinherentlymore mentwithourexperimentalvalue. Naively, sucharesult
interesting than a [1-10] field, as no ψ state can fully can be interpreted as the ψ selection in Er Ti O be-
2 2 2 2 7
minimize the Zeeman energy. Thus, there is competi- ing largely dominated by order-by-disorder effects, with
7
virtualcrystal-fieldexcitationscontributingrelativelylit-
tle to the ψ selection. However, the critical field for 1 (a) H || [1-10] ψ2
2 0.8 ψ
the ψ to ψ spin-flop transition can only be indirectly 3
2 3
0.6
modeled, introducing a degree of uncertainty. Further ak
analysis, perhaps via numerical methods, may allow a Pe 0.4
g
more accurate comparison between the strength of the g 0.2
a
0 T order-by-disorder six-fold clock term and the experi- Br 0
c
mentaldata. Nonetheless,inprinciple,ourmeasurement neti 1 (b) H || [001]
of the 0 T six-fold clock term should provide important ag 0.8
M
information that would allow the ground state selection 0) 0.6
in Er Ti O to be definitively understood. 2
2 2 7 2 0.4
y of ( 0.2
nsit 0
C. Domain Selection and Reorientation in a [111] e
Magnetic Field d Int 1 (c) H || [111]
e 0.8
er
The Zeeman coupling for a field along the [111] direc- att 0.6
c
tion is predicted to induce a combination of a three-fold S 0.4
andsix-foldclocktermsand,hence,thedomainbehavior 0.2
of Er2Ti2O7 in a [111] field is expected to be rich [17]. 0
The three-fold clock term selects the ψ2 states with an- 0 π/2 π 3π/2 2π
gles corresponding to ±π/3 and π. However, the six-fold XY local angle, θ
clocktermisnotthesameastheonethatselectsthe0T
ψ states. Instead, this [111] six-fold Zeeman clock term FIG.6: Thescatteredintensityatthe(220)Braggpositionin
2
selects states which are rotated with respect to both the Er2Ti2O7 asafunctionoftheangle,θ,intheXYlocalplane.
The red dots represent the six ψ domains and the blue dots
ψ and ψ states. This term then competes with both 2
2 3 represent the six ψ domains. In (a) the two ψ domains cir-
the three-fold Zeeman clock term and also against the 3 2
cled in black are the ones selected when a magnetic field of
0 T six-fold clock term that favors the ψ states.
2 order 0.1 T is applied along the [1-10] direction. In (b) the
At low field, it is predicted that the domain selec- four domains circled in black are the four ψ states immedi-
2
tion behavior should be dominated by the combination atelyselectedina[001]field. Thesestatesthencanttowards
of both the three-fold Zeeman clock term and the 0 T the two ψ3 states indicated by the black squares, which are
selected by a 0.18 ± 0.05 T [001] magnetic field. In (c) the
six-fold clock term. First, at very low fields, the three
three ψ domains circled in black are the ones selected for
domains with an angle of ±π/3 and π should be selected 2
magneticfieldsupto0.15Tappliedalongthe[111]direction.
(see black circles on Fig. 6(c)), which would result in no
At higher fields, these domains are predicted to split by an
change in the (220) intensity. Experimentally, we refer angle, θ, as indicated by the arrows. Our measurement in
backtoourmeasurementshowninFig.3(f),whichshows a [111] field measured the (2-20) Bragg peak, which is sym-
that the elastic scattering is constant up to 0.15 T. This metricallyequivalentto(220)andgivestheidenticalresultas
is then consistent with the theoretical prediction that shown in panel (c)
the three indicated ψ states are selected for small [111]
2
fields. Moreover, our data is in good agreement with the
theoretical prediction that a weak [111] field induces an ±π/3 and π domains. These transitions would be diffi-
emergent three-fold clock term. culttoverifyusingunpolarizedneutronscattering,asthe
Upon further increasing the [111] field, it is predicted domainsplittingbyanangle,θwouldhavezeroneteffect
that Er Ti O should experience two additional domain ontheintensityof(220). Thus,fora[111]field,the(220)
2 2 7
transitions prior to entering its classical field polarized Bragg peak is predicted to have no changes to its inten-
state. Theoriginofthesetransitionsistheemergentsix- sity up until the transition to the field polarized state.
fold Zeeman clock term. The first of these transitions is The predicted (220) behavior calculated by Maryasin et
theoretically predicted to occur at H =0.16 T and re- al. [17] is indicated by the red line in Fig. 3(f) and it
c1
sultsintheformationofsixnewdomains. Thesesixnew is apparent that our experimental observations are not
domains are related to the previously selected ±π/3 and fully consistent with the predicted scenario, as the in-
π domains,butwithasplittingangleof±θ(seeblackar- tensity of (220) is observed to substantially vary above
rows in Fig. 6(c)). This splitting angle, θ, is predicted to 0.15 ± 0.03 T. We note, however, the predicted critical
have the same value for all domains and should increase fieldsforthesetransitions,H =0.16TandH =0.4T
c1 c2
as a function of the applied field. At still higher fields, do appear to be meaningful in Er Ti O : the first, H ,
2 2 7 c1
H = 0.4 T, a second transition decreases the splitting corresponds with the observed increase in elastic inten-
c2
angle back to zero, returning the system to a state with sity at (220) by a factor of 1.75 and the second, H ,
c2
8
corresponds quite well with the field at which the inten- [1-10]field, wherethespinsaremaximallyperpendicular
sity flattens, 0.4 ± 0.03 T. to the scattering vector, the quasi Goldstone-mode ex-
To account for the observed intensity gain of (220) citations must necessarily have their intensity reduced,
above 0.15 T, we propose a scenario of non-equiprobable as the spins will become more parallel to Q and will
domain distribution or inequivalent splitting angles become less visible to neutron scattering. The opposite
(Fig. 6(c)). Such scenarios would explain the increase is true for a [001] field, where the spins are maximally
in the elastic scattering at (220), but would require an parallel to the scattering vector and must excite into a
additionalZeemanclocktermthatwouldfavorstheπdo- more perpendicular orientation. The effect of a field on
main over the ±π/3 domains. In the theoretical work of the scattered intensity of the quasi Goldstone-mode are
Maryasin et al., no such term is predicted for a perfectly thus, another direct signature of the domains effects.
alignedsample[17]. OnepossibleoriginofsuchaZeeman Moreover, the field dependence of the inelastic and elas-
clock term is a slight misalignment of the field along the ticscatteringinEr Ti O areconsistentwitheachother.
2 2 7
[111] direction. While we can place an upper bound of
2◦ on the error in our alignment, it is worth noting that
very small misalignments in a [111] field are known to
be enhanced by demagnetization effects in spin ice [28]. V. CONCLUSIONS
However, in contrast to the spin ice case, Er Ti O is a
2 2 7
antiferromagnet where such demagnetization effects are We have performed comprehensive time-of-flight neu-
naivelyexpectedtobefarlessimportant. Thus, thepre- tron scattering measurements on single crystals of
cise domain distributions and orientations occurring for Er Ti O with a magnetic field applied along the [001]
2 2 7
fields between 0.15 and 0.4 T in a [111] field remain an and[111]directions. Forfieldssmallerthan1T,thefield
open question at present. induced effects can be attributed to domain selection.
The zero field state of Er Ti O assumes the ψ antifer-
2 2 7 2
romagnetic structure, which is composed of six equally
D. Effect of Domain Selection on the Quasi probabledomains. Forsmallfieldsappliedalongthe[001]
Goldstone-Mode Excitations direction, we observe a dramatic decrease of the (220)
magnetic Bragg peak intensity, which agrees very well
Finally, it is interesting to comment on the low with the predicted transition from the six ψ domains to
2
field behavior of the quasi Goldstone-mode excitations twoψ domains. Forthe[111]fielddirection, weobserve
3
centeredon(220). Inasmall[1-10]field, aswediscussed thattheelasticscatteringat(220)isindependentoffield
in the Results section III.B., the quasi Goldstone-mode forlowfields,consistentwithanemergentthree-foldZee-
excitationsdisappearatthisparticularorderingwavevec- man clock term. Further increasing the field results in a
tor. This is best observed by comparing the inelastic large enhancement of the (220) Bragg peak intensity, in-
spectra of 0 T and 0.5 T in Fig. 4(a) and Fig. 5(a). The consistent with the predicted domain selection scenario
disappearance of the quasi Goldstone-mode excitations andhintingatanevenricherphasebehavior. Lastly,our
is concomitant with the increase of the elastic scattering experiment provides a measure of the zero field six-fold
at (220) (Fig. 3(b)). The exact opposite behavior is clock term strength, 0.18 ± 0.05 T. This result should
observed for a field along the [001] direction, where prove useful in establishing a complete understanding of
we observe that the quasi Goldstone-mode excitations the mechanism of ground state selection in Er Ti O .
2 2 7
increase for small fields (Fig. 4(b) and Fig. 5(b)), while The work presented here experimentally confirms the
the elastic scattering (Fig. 3(d)) decreases for the same rich and exotic magnetothermodynamic behavior pre-
field range. Thus, there is a clear trade off between dicted to exist in the XY pyrochlore antiferromagnet
the intensities of the elastic scattering and the quasi Er Ti O , or any other pyrochlore magnet ordering into
2 2 7
Goldstone-mode scattering at the (220) magnetic Bragg the Γ manifold. We demonstrate that, depending on
5
peak while transitioning into states with different XY the field direction, a combination of two-, three-, and
local angles. To understand this observation, we suggest possibly six-fold Zeeman clock terms emerge and com-
that the intensity of the quasi Goldstone-mode at (220) pete with the six-fold clock term present in zero field.
goes like I ∝ sin2(θ). This can be understood by first We anticipate this work will help determine the ground
pointing out that for magnetic neutron scattering, it is state selection and field effects in other topical Γ py-
5
the component of the moment perpendicular to Q that rochlores, such as NaCaCo F [29], Er Ge O [30] and
2 7 2 2 7
couples to the neutron (ie. in this case perpendicular to Yb Ge O [30, 31].
2 2 7
(220)). The domains for which the (220) elastic scatter- We thank M. Zhitomirsky and R. Moessner for very
ing is maximized have their spin directions maximally useful discussions and for providing us the theoretical
perpendicular to the (220) direction, and vice-versa. curvesofthefielddependenceofthe(220)Braggpeakin-
However, for low fields, the spins remain constrained to tensityshowninFig.3(b,f). WealsowishtothankYegor
lie within the local XY plane. Thus, in the case of a Vekhov for his technical support with the dilution fridge
9

used in the neutron experiment. The neutron scattering
data were reduced and analyzed using DAVE software
package [32]. The NIST Center for Neutron Research is
supported in part by the National Science Foundation
under Agreement No. DMR-094472. Works at McMas-
ter University was supported by the National Sciences
and Engineering Research Council of Canada (NSERC).
Appendix A: Field effects on the Lorentzian
contribution of the (220) magnetic Bragg peak
Asdiscussedinthemanuscript,elasticcutsofthe(220)
magnetic Bragg peak have been carefully modeled with
a two component fit for all field strengths and orienta-
tions. AnexampleofonesuchfitisshowninFig.7(a)for
the [111] sample at 0 T. The Gaussian component of the
scatteringoriginatesfromthenuclearandmagneticlong-
range order and its field dependence is thoroughly dis-
cussed in the manuscript. The second component of the FIG. 7: (a) A typical fit of the scattering at the (220)
Bragg peak position using both a Gaussian function and a
scattering at (220) is captured by a Lorentzian function,
Lorentzian function, to capture the elastic and inelastic con-
andaccountsfortheinelasticpartofthescatteringdueto
tributions to the scattering, respectively. The field depen-
thepartialintegrationoverthequasiGoldstone-modeex-
dence of the Lorentzian contribution of the scattering for a
citations. The Lorentzian contribution of the scattering (b)[1-10]magneticfield,(c)[001]magneticfield,and(d)[111]
issomewhatinterestingasitprovidesinformationonthe magneticfield. ThefielddependenceoftheLorentziancontri-
verylowenergyinelasticfielddependence. Thefieldevo- butiontrackstheintensityofthelowenergyquasiGoldstone-
lution of the integrated intensity of the Lorentzian com- mode excitations.
ponent is shown for all field directions in Fig. 7(b,c,d).
For the [1-10] field direction, the Lorentzian part of
moves out of our elastic window. This shifting of the in-
the scattering reaches zero intensity for a field of 0.5 T
elastic intensity to higher energies signals that the spin
(Fig.7(b)). Thisdecreaseofscatteringisconsistentwith
wave gap of the quasi Goldstone-mode is growing from
the complete removal of the quasi Goldstone-mode exci-
0.1 T up to 1 T. Above 1 T, the Lorentzian part of the
tations (Fig. 4(a)). The Lorentzian contribution of the
scattering completely disappears as the (220) magnetic
scattering maintains zero intensity up to 1.5 T, at which
Braggpeakhaszerointensityinthefieldpolarizedstate.
point a clear and abrupt increase is observed. This orig-
Lastly, we turn our attention to the field evolution of
inates from the softening of the spin wave excitations
theLorentziancomponentina[111]fieldwhichisshown
towards the elastic line (Fig. 4(a) Fig. 5(a)) and, thus,
in Fig. 7(d). Similar to the low field dependence of the
arepickedupbyourintegrationovertheelasticchannel.
elasticscattering(Fig.3(f)),weobservethattheinelastic
Above1.5T,noLorentziancontributionofthescattering
intensityisflatupto0.15T.Above0.15T,theintensity
is observed in a [1-10] field.
falls off abruptly up to 1 T. This decrease of scatter-
For the [001] field direction, the Lorentzian contri-
ing is consistent with the reduced intensity of the quasi
bution increases dramatically at 0.1 T, due to the en-
Goldstone-modeexcitations(Fig.4(c)andFig.5(c)). As
hancement of the scattering from the quasi Goldstone-
was the case for the [1-10] field, for 1.5 T, we observe
modeexcitations(Fig.7(c)). Indeed,asseeninFig.4(b)
a large enhancement of the Lorentzian intensity. This
and Fig. 5(b), the quasi Goldstone-mode excitations are
originate from the softening of the spin wave excitations
clearly enhanced upon application of fields up to 0.5 T.
towards the elastic line. This feature is best observed
Referring once again to Fig. 7(c), we can see that above
by looking at the 1.5 T energy slice shown in Fig. 4(c).
0.1T,theinelastic(Lorentzian)contributionsteadilyde-
Above 1.5 T, the spin wave excitations are pushed to
creases, reaching zero intensity for fields above 0.75 T.
higher energy and the Lorentzian contribution of the
This effect can also be explained by examining the en-
scattering remains at zero intensity.
ergy cuts of Fig. 5(b). Comparing the data sets between
0.1 T and 1 T, we see that the inelastic intensity shifts
to progressively higher energies with increasing field. As
weareintegratingoverourelasticresolutioninFig.7(c),
from −0.1 to 0.1 meV, this integration picks up less of ∗ Electronic address: [email protected]
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