Low-energy spin dynamics of orthoferrites AFeO3(A = Y, La, Bi)

Kisoo Park1,2, Hasung Sim1,2, Jonathan C. Leiner1,2,Yoshiyuki Yoshida3, Jaehong Jeong1,2,4, Shin-ichiro Yano5,Jason Gardner5,6, Philippe Bourges4, Milan Klicpera7,8,Vladimı́r Sechovský7, Martin Boehm8 and Je-Geun Park1,2

1 Center for Correlated Electron Systems, Institute for Basic Science (IBS),Seoul 08826, Republic of Korea2 Department of Physics and Astronomy, Seoul National University, Seoul08826, Republic of Korea3 National Institute of Advanced Industrial Science and Technologh (AIST),1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan4 Laboratoire Léon Brillouin, CEA, CNRS, Université Paris-Saclay, CEA Saclay,F-91191 Gif-sur-Yvette Cedex, France5 Neutron Group, National Synchrotron Radiation Research Center, Hsinchu30077, Taiwan6 Center for Condensed Matter Sciences, National Taiwan University, Taipei10617, Taiwan7 Faculty of Mathematics and Physics, Department of Condensed MatterPhysics, Charles University, Ke Karlovu 5, 121 16 Praha 2, Czech Republic8Institut Laue-Langevin, 71 avenue des Martyrs, CS 20156, 38042 GrenobleCedex 9, France

E-mail: jgpark10@snu.ac.kr

March 2018

Abstract. YFeO3 and LaFeO3 are members of the rare-earth orthoferritesfamily with Pbnm space group. Using inelastic neutron scattering, the low-energyspin excitations have been measured around magnetic Brillouin zone center.Splitting of magnon branches and finite magnon gaps (∼2 meV) are observed forboth compounds, where the Dzyaloshinsky-Moriya interactions account for mostof this gap with some additional contribution from single-ion anisotropy. We alsomake comparisons with multiferroic BiFeO3 (R3c space group), in which similarbehavior was observed. By taking into account all relevant local Dzyaloshinsky-Moriya interactions, our analysis allows for the precise determination of allexperimentally observed parameters in the spin-Hamiltonian. We find thatdifferent properties of the Pbnm and R3c space group lead to the stabilization ofa spin cycloid structure in the latter case but not in the former, which explains thedifference in the levels of complexity of magnon band structures for the respectivecompounds.

Keywords: Ferrites, Multiferroics, Inelastic neutron scattering, Dzyaloshinskii-Moriyainteraction, weak ferromagnetism

Submitted to: J. Phys.: Condens. Matter

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Low-energy spin dynamics of orthoferrites AFeO3 (A = Y, La, Bi) 2

1. Introduction

Magneto-electric (ME) multiferroic materials, in whichboth magnetic and ferroelectric ordering coexist, haveattracted much attention due to the tunable magneticproperties via electric field or vice versa. Such materi-als also present the possibility of various applicationsin recording device technology or spintronics [1, 2, 3].While searching for appropriate candidates is far fromtrivial, one may consider compounds with weak fer-romagnetism (wFM) where the reversal of wFM by180◦ using electric field has been predicted theoreti-cally [4]. In many cases, the microscopic mechanism ofwFM is either Dzyaloshinsky-Moriya (DM) interactionor single-ion anisotropy (SIA) [5, 6, 7]. In this regard,accurately measuring the values of such quantities inreal materials is of considerable importance for futureapplications.

The rare-earth orthoferrites RFeO3 are one ofmost promising model systems in this regard. TheFe3+ ions in all of the RFeO3 family undergo anantiferromagnetic transition with TN ranging from623 K in R=Lu to 738 K in R=La. These hightransition temperatures are due to a strong nearest-neighbor exchange interaction (J > 4 meV) alongthe Fe−O−Fe bond and the large magnetic momentof Fe3+ (S = 5/2). Most perovskites of ABO3-typeexhibit a cubic Pm3̄m structure at high temperature,and a structural transition occurs upon cooling whichlowers the symmetry via tilting of edge-shared BO6octahedra. RFeO3 adopts the Pbnm space group atthis structural transition, the most frequent structureamong the perovskites. Such octahedra tilting toPbnm symmetry can be described by Glazer notation:a−a−c+ [8]. Since this structure does not break spaceinversion symmetry (i.e. Pbnm is centrosymmetric),no net polarization in RFeO3 is expected.

In the case of RFeO3, the tilting of FeO6octahedra is the origin of local DM interaction in thiscompound (see figure 1). Competition between DMand exchange interactions results in canting magneticmoments [9]. Below TN , all RFeO3 adopt a cantedantiferromagnetic ground state Γ4(Ga, Ab, Fc) withbasic G-type antiferromagnetism along the a-axis,weak antiferromagnetism along the b-axis, and weakferromagnetism along the c-axis as shown in figure 1(a).Such weak canted magnetic moments were extensivelystudied both theoretically and experimentally [10, 11].A. S. Moskvin and E. V. Sinitsyn derived a simpleformula connecting the canting of magnetic momentand the crystal properties (unit cell parameter,position of oxygen and the bond length), deducing arelation between the Ay and Fz [12]. This theoreticalprediction was confirmed for several orthoferrites bythe polarized neutron diffraction [13, 14, 15]. ForYFeO3, calculated value of Ay/Fz = 1.1 is consistent

c

ba

Y/La

Fe

O

(a) (b)

𝑆1

𝑆2𝑆3

𝑆4

𝐷34 −𝛼𝑐 , 𝛽𝑐 , 0

𝐷41 −𝛼𝑎𝑏 , −𝛽𝑎𝑏 , 𝛾𝑎𝑏

𝐷41′ 𝛼𝑎𝑏 , −𝛽𝑎𝑏 , 𝛾𝑎𝑏

𝐷32 −𝛼𝑎𝑏 , 𝛽𝑎𝑏 , 𝛾𝑎𝑏

𝐷32′ 𝛼𝑎𝑏 , 𝛽𝑎𝑏 , 𝛾𝑎𝑏

𝐽′

𝐽𝑎𝑏

𝐽𝑐

𝑆1 𝑆2𝑆3

𝑆4

Figure 1. (Color online) (a) Magnetic structure of YFeO3 andLaFeO3. Exchange interactions bewteen nearest-neighbor bondsalong ab-plane (Jab) and c-axis (Jc), next-nearest neighborbonds (J ′) are shown by the dashed arrows, respectively. (b)Local DM vectors (arrows) for nearest neighbors on distortedoctahedra.

with the experimental results within errorbars. It isworth noting that in case of RFeO3 with magneticrare-earth ions, there is a magnetic ordering ofR3+ at low temperature and a spin reorientationtransition of Fe3+ at intermediate temperatures dueto the interaction between R3+ and Fe3+ ions. Suchadditional interactions between the two magneticions sometimes induces multiferroicity below the spinreorientation transition temperature, and often resultsin the rotation of Fe3+ ions by exchange-strictionmechanism [16, 17, 18].

BiFeO3 is the only example that is well-establishedto exhibit multiferroicity above room temperature.BiFeO3 shares several characteristics with RFeO3: ithas the similar exchange interaction and the very highantiferromagnetic transition temperature TN at ∼650K [19]. However, there are also clear contrasts betweenthese two materials such as the distinct rotation ofFeO6 octahedra, much of which is due to the lonepair of Bi breaking the inversion symmetry for BiFeO3unlike the other centrosymmetric RFeO3. BiFeO3has the non-centrosymmetric space group R3c comingfrom the Glazer tilting a−a−a−. BiFeO3 exhibitsa large polarization with a ferroelectric transition atTc=1100 K [20]. Below TN , an incommensuratespin cycloidal magnetic structure develops along the[1 1 0] direction with an extremely long period of620 Å and is superimposed on the simple G-typeantiferromagnetism [21]. It was also reported to havea negative magnetostrictive magnetoelectric couplingat TN [22]. Small angle neutron scattering (SANS)experiments revealed a spin density wave (SDW)fluctuation, which is perpendicular to the spin cycloid[23]. The local wFM moment made by this fluctuationis cancelled out over the whole cycloid, giving no wFMin bulk BiFeO3.

The spin-Hamiltonian of BiFeO3 has been ex-tremely well studied both theoretically and experimen-tally throughout many studies [24, 25, 26, 27, 28]. Re-

Low-energy spin dynamics of orthoferrites AFeO3 (A = Y, La, Bi) 3

cent study on the magnetic excitation spectra over thefull Brillouin zone using inelastic neutron scattering(INS) measurements determined the values for the twoexchange interactions and the Dzyaloshinskii-Moriya(DM) interaction [25]. Subsequently, a detailed exam-ination was done on the low-energy region with theobservation of the unique island-like feature at 1 meV.Separately, this can also be identified as the peak-and-valley feature in the constant Q-cut graph at the mag-netic zone center [26, 27]. By employing the full spinHamiltonian in spin wave calculations, it was furtherdetermined that this feature originates from the inter-play of the DM interaction and the easy-axis anisotropy[27].

The rare-earth orthoferrites have also beenpreviously characterized in the literature, includingstudies on the spin waves of RFeO3 with INS [29, 30,31, 32, 33] and Raman spectroscopy [34, 35, 36]. Muchof the focus in the INS studies was concentrated on thehigh energy transfer region of the excitation spectrato determine the structural and magnetic interactionstrengths. For LaFeO3, only powder INS spectrawas reported that confirmed Heisenberg type nearest-neighbor exchange interactions between Fe3+ ions [32].For YFeO3, a recent INS study successfully measuredthe overall shape of magnon dispersion up to ∼70meV and deduced the best fit parameters includingthe nearest- and next nearest-exchange interactionsJ1 and J2, DM interactions, and SIA [33]. Inaddition, the low-energy transfer region at the Brillouinzone center was examined by Raman spectroscopy.These Raman measurements for YFeO3 determinedthe magnon peaks around ∼1.4 and 2.2 meV at theΓ point [34]. Using these data, they also determinedthe parameters of the spin Hamiltonian of YFeO3.However, the model Hamiltonian used in the abovestudies needs to be improved as it does not captureall the salient details of Pbnm symmetry, in particularthe local DM vectors and their relation with the cantedferromagnetic moment. We note that local DM vectorsare present even for centrosymmetric space group likePbnm of RFeO3, and it is rather poorly understoodhow this local DM vectors affect the spin waves.

To understand the differences between these twocompounds and the role of local DM vectors, itis necessary to quantitatively determine their fullspin Hamiltonian. In this work, we have carriedout comprehensive studies on the low-energy magnonexcitations of YFeO3 and LaFeO3 since this is whereeffects of DM interaction and SIA are expected tomanifest most strongly. We also collected new dataof the low-energy spin waves of BiFeO3 focusing onhigher momentum resolution. Note that we havepurposely selected the nonmagnetic rare-earth YFeO3and LaFeO3 orthoferrites in order to focus directly

on the magnetism of Fe3+. Based on the allowedform of the DM interactions in the Pbnm symmetry,we have quantified the parameters of the full spinHamiltonian for YFeO3 and LaFeO3, and reproducedtwo characteristic features observed in the low-energymagnetic excitation spectra: (1) a finite spin wavegap and (2) splitting of two magnon branches atthe zone center. Also, two additional shoulders inconstant energy cuts of BiFeO3 have been identified,demonstrating the more complex nature of the magnonbranches in comparison to the other orthoferrites.

2. Experimental Details

Single crystals of YFeO3 and LaFeO3 with masses of1.52 and 1.41 g respectively were grown with floatingzone furnaces. INS experiments were performedulitizing the cold-neutron triple axis spectrometerSIKA [37] at the Australian Nuclear Science andTechnology Organisation (ANSTO). Samples weremounted with their orthorhombic b∗-axis vertical, suchthat the wave vectors of the observed spin waves wereall confined to the a∗−c∗ plane. Based on the reflectionconditions of magnetic Bragg peaks for the Pbnm spacegroup, all constant-Q energy scans were carried outalong the [H 0 0] direction centered on Q = (1 0 1).The final neutron energy was fixed at 5 meV giving afull width at half maximum (FWHM) energy resolutionof 0.106 meV at the elastic position. A beam collimatorconfiguration of 40′−40′−60′−40′ was used to obtainoptimized beam intensity and resolution. A cooledpolycrystalline berylium filter was installed to removethe higher-order contamination of the scattered beam.Data were collected at 300 K without a cryostat, andthen at 1.5 K with an orange cryostat.

For BiFeO3, the INS experiments were done withtwo cold-neutron triple axis spectrometers: 4F2 atLaboratoire Leon Brillouin (LLB) and ThALES atInstitute Laue-Langevin (ILL). The data obtained atLLB have already been presented by Jeong et. al. [27]and reproduced here for comparison and subsequentdiscussion. In all measurements with 4F2, eight co-aligned single crystals of total mass 1.6 g with 3◦

mosaicity were used. To achieve better momentumresolution, one single crystal with mass of 0.58 g wasused in ThALES experiment. Similar with RFeO3,BiFeO3 samples were aligned in the a

∗−c∗ plane. Using4F2, energy scans along the [H 0 0] direction centeredon Q = (1 0 -1) at T = 16 and 270 K with fixedkf = 1.2Å

−1. In additional measurements with theThALES instrument, we have measured the constant-energy (E = 3 meV) cut along the [H 0 0] directioncentered on Q = (1 0 -1) at T = 270 K.

Low-energy spin dynamics of orthoferrites AFeO3 (A = Y, La, Bi) 4

0 . 0

0 . 5

1 . 0

1 . 5

( d ) L a F e O 3 , T = 1 . 5 K

( b ) Y F e O 3 , T = 1 . 5 K�''

(a.u.)

H = 1 H = 1 . 0 1 H = 1 . 0 2 H = 1 . 0 3

( a ) Y F e O 3 , T = 3 0 0 K

( c ) L a F e O 3 , T = 3 0 0 K

H = 1 H = 1 . 0 1 H = 1 . 0 2 H = 1 . 0 3

0 2 4 60 . 0

0 . 5

1 . 0

1 . 5

�'' (a.

u.)

E n e r g y ( m e V )

H = 1 H = 1 . 0 1 H = 1 . 0 2 H = 1 . 0 3

0 2 4 6E n e r g y ( m e V )

H = 1 H = 1 . 0 1 H = 1 . 0 2 H = 1 . 0 3

Figure 2. (Color online) Constant Q-cuts along the [H 00] direction centered at Q = (1 0 1) of (a,b) YFeO3 and (c,d)LaFeO3 at T = 300 and 1.5 K. Symbols represent the data pointsand solid lines denote the convoluted intensity I(Q,ω) calculatedfrom our simulation as discussed in the text.

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

( d ) L a F e O 3 , E = 5 m e V

( b ) Y F e O 3 , E = 5 m e V

�'' (a.

u.)

( a ) Y F e O 3 , E = 2 m e V

( c ) L a F e O 3 , E = 2 m e V

0 . 9 8 1 . 0 0 1 . 0 20 . 0

0 . 2

0 . 4

0 . 6

0 . 8

�'' (a.

u.)

( H , 0 , 1 ) ( r . l . u . )0 . 9 8 1 . 0 0 1 . 0 2

( H , 0 , 1 ) ( r . l . u . )

Figure 3. Constant E-cuts at E = 2 and 5 meV along the[H 0 0] direction centered at Q = (1 0 1) of (a,b) YFeO3 and(c,d) LaFeO3 at T = 300 K. Symbols represent the data pointsand solid lines denote the convoluted intensity I(Q,ω) calculatedfrom our simulation as discussed in the text.

3. Results and Discussion

Figure 2 shows scans in energy transfer at various Qpoints along the [H 0 0] direction centered at Q =(1 0 1) of YFeO3 and LaFeO3 for T = 300 and 1.5K. After corrected for the Bose factor, the measuredneutron intensities are proportional to the dynamicsusceptibility Im[χ(Q,ω)]. For both compounds, thedefining features in the constant-Q cuts are as follows:(1) a finite spin wave gap of E ∼1 meV (YFeO3) and2 meV (LaFeO3) and (2) two distinct peaks directlyabove the gap, although the valleys between the twopeaks are quite small. The two peaks are, as expected,most distinguishable at Q = (1 0 1), signifying that themagnon branches are split at the magnetic Brillouin

zone center. Figure 3 denotes the constant-energytransfer graphs of YFeO3 and LaFeO3 for T = 300 K.One can see that the magnetic signals at low energyare separated as two peaks as the energy transferincreases, implying the V-shaped dispersion of themagnetic excitation of YFeO3 and LaFeO3.

In order to fully explain the low-energy magneticexcitations, we employ a minimal spin Hamiltonian ofRFeO3 to model the experimental data:

H = Jc∑

along c

Si · Sj + Jab∑

ab plane

Si · Sj

+ J ′∑〈〈ij〉〉

Si · Sj +∑〈ij〉

Dij · Si × Sj (1)

+Ka∑i

(Sxi )2

+Kc∑i

(Szi )2,

where Jc and Jab represent the nearest-neighborexchange constants along the c-axis and the ab plane,respectively. In the previous INS study on YFeO3 [33],these Jc and Jab were set as same value J1. However,we note that the difference between Jc and Jab canreach up to 10 % due to Bloch’s rule [38], especiallyin the case of YFeO3. J

′ denotes the exchangeconstant along the next-nearest neighbor bonds (seefigure 1(a)). The fourth term represents the DMinteractions defined on the Fe(i)-O-Fe(j) bonds withthe antisymmetric relation: (Dij = −Dji). Transitionions having a 3d5 configuration such as Fe3+ lead toA1g orbital symmetry. Therefore, we may assumethat the DM interaction of ferrites can be given bya microscopically derived form (Dij ∝ x̂i× x̂j) [39, 40],where x̂i is the unit vector connecting i-th Fe atom andoxygen atom between i-th and j-th Fe atoms. Thismeans that in the Pbnm structure all DM interactionsbetween two adjacent iron atoms may be characterizedby five parameters: αab, βab, γab, αc, βc [41], as shownin figure 1(b). The density functional theory (DFT)calculation on LaFeO3 [42] shows good agreement withthe DM vectors obtained from our structural analysis,supporting this assumption.

Normalized values of the local DM vectors ofYFeO3 and LaFeO3 are shown in table 1. We notethat the in-plane DM vectors defined in different basalplanes, e.g. D41 and D32, are different along the b-axis. The result of combining all contributions ofadjacent ions is that every Fe3+ ion feels a different DMinteraction, therefore global DM interactions cannot beas expected defined in this space group. This is anassumption contrary to those used in previous studieson YFeO3 [33, 34].

The last two terms of equation 1 denote the easy-axis (Ka, Kc

Low-energy spin dynamics of orthoferrites AFeO3 (A = Y, La, Bi) 5

0

2

4

6

8E

nerg

y (m

eV)

(a) YFeO3, T = 300 K (b) LaFeO3, T = 300 K E = 3 meV(c) BiFeO3T = 270 K

0.96 0.98 1.00 1.02 1.040

2

4

6

8

Ene

rgy

(meV

)

(H, 0, 1) (r.l.u.)

(d) YFeO3, simulation

0.96 0.98 1.00 1.02 1.04(H, 0, 1) (r.l.u.)

(e) LaFeO3, simulation

0.96 0.98 1.00 1.02 1.04(H, 0, -1) (r.l.u.)

(f) BiFeO3, simulation

0.0

0.5

1.0

1.5

�'' (

a.u.

)

0

1

2

�'' (

a.u.

)

Figure 4. (Color online) (a-c) Contour plots of the INS intensity of (a) YFeO3 at T = 300 K, (b) LaFeO3 at T = 300 K and (c)BiFeO3 at T = 270 K along the [H 0 0] direction in the reciprocal space. (d-f) theoretically calculated single-magnon dispersioncurve and Im[χ(Q,ω)] of (d) YFeO3, (e) LaFeO3 and (f) BiFeO3. Experimental data and calculation in (c) and (f) were taken fromreference [27]. The upper part of (c) denotes the constant-energy (E = 3 meV) cut along the [H 0 0] direction centered at Q = (10 -1) for BiFeO3 taken from the ThALES spectrometer. Individual Gaussian peaks of single magnon branches are shown as dashedlines with filled area, while the solid line denotes the total sum of all peaks.

Table 1. Normalized components of local DM vectors of YFeO3and LaFeO3.

αab βab γab αc βcYFeO3 0.517 0.488 0.703 0.346 0.938LaFeO3 0.554 0.553 0.623 0.191 0.982

spin wave theory, SIA is the origin of the spin wavegap at the Brillouin zone center. It is worth notingthat the most generalized form of the spin Hamiltonianalso includes the symmetric anisotropic exchangeinteraction, i.e. two-ion anisotropy (TIA). SuchTIA terms are formulated as the form

∑ij SiΩijSj ,

where Ωij denotes 3 × 3 symmetric matrix. This ischaracterized by eight different parameters related toits Pbnm symmetry. This anisotropy, however, seemsto be small with the order of D2/J [43], and wouldadd unnecessarily too many parameters to our modelHamiltonian. The TIA mostly affects the spin wavegap at the zone center, like the SIA. In that sense,this TIA can be neglected and therefore will not bediscussed further in this study.

After combining all contributions from the oxygenenvironments, the four-sublattice magnetic groundstate Γ4(Ga, Ab, Fc) of the RFeO3 can be stabilized[42]. In spherical coordinates, the four spins can bedefined using two spin canting angles θ and φ, whichare related to the weak ferro- and antiferro- magnetic

moment, respectively.

S1 = S (−cosθcosφ,−cosθsinφ, sinθ)S2 = S (cosθcosφ, cosθsinφ, sinθ)

S3 = S (−cosθcosφ, cosθsinφ, sinθ)S4 = S (cosθcosφ,−cosθsinφ, sinθ)

(2)

Since the spin cantings are very small (∼ 0.5◦) forRFeO3, we can ignore terms higher than second orderwith respect to the spin-orbit coupling λSO to obtainthe relationship between spin canting angles and thespin Hamiltonian parameters from the ground stateenergy [13]:

θ =2βab + βc

4Jab + 2Jc +Kc −Ka,

φ = − 2γab4Jab − 8J ′ −Ka

(3)

Using this Hamiltonian of RFeO3, we tried tofind the best fit parameters that reproduce theexperimental result well. First, an initial set ofparameters was chosen under several constrainingconditions. As the Hamiltonian contains manyparameters: Jc, Jab, J

′, Dab, Dc, Ka and Kc, utilizingall the reasonable initial and constraining conditionsis important for determining a reliable set of best fitparameters. Therefore, starting with the previouslyreported exchange coupling constants J1 = 4.77 and J2= 0.21 meV derived from high energy INS experiment[33], Jc, Jab and J

′ were refined. Since only the J1

Low-energy spin dynamics of orthoferrites AFeO3 (A = Y, La, Bi) 6

0 2 4 60 . 0

0 . 5

1 . 0

1 . 5

�'' (ar

b. un

it.)

E n e r g y ( m e V )Figure 5. (Color online) Constant Q-cut (black rectangle) atthe magnetic Brillouin zone center compared with our simulation(line) of BiFeO3. Experimental data and calculation were takenfrom reference [27].

value of LaFeO3 has been previously reported [32], wemade the assumption that the J1/J2 ratio of LaFeO3 issimilar with that of YFeO3. This assumption combinedwith the ratio of TN of both compounds, yields Jc =Jab = 5.47 and J

′ = 0.24 meV for LaFeO3. We alsoused in our analysis the canting angle θ as derivedfrom polarized neutron diffraction results [13] andmagnetization measurements along the c-axis [44]. Toobtain the consistency between the spin canting anglesand the spin Hamiltonian parameters, equation 3 wasused as one of the constraint conditions.

Secondly, with the chosen initial parametersfitting was performed by a bounded non-linear leastsquares fit to the experimental data set. Due tothe presence of the constraint condition (equation3), fmincon programming solver implemented inMATLAB was used. During the non-linear fit, thetheoretical magnon dispersion curve and dynamicstructure factor S(Q, ω) have been calculated. Wenote that the derivation of the analytic form of thedispersion is not easy as the size of Hamiltonian matrixis 8 × 8. We used SpinW software package [45]to diagonalize the spin Hamiltonian in the Holstein-Primakoff approximation.

Since the neutron intensity obtained from thetriple axis spectrometer is convoluted with theinstrumental 4D resolution ellipsoid in the momentum-energy space, the theoretically derived dynamicstructure factor should also be convoluted withthe resolution ellipsoid for direct comparison withexperimental data. The total INS intensity measuredby the triple axis spectrometer is given by [46]:

I(Q0, ω0) ≈ R0∫d3QdωS(Q,ω)

× exp[−12

∆℘iMij(Q0, ω0)∆℘j ],

(4)

where Q0 = ki − kf represents the momentum transferto the sample, h̄ω = Ei − Ef is the energy transfer,

∆℘ ≡ (Q − Q0, h̄(ω − ω0)), and M is a 4 × 4 matrixdefining a 4-dimensional resolution ellipsoid. Based onthe geometry of the SIKA beamline and information ofthe sample, M matrices were calculated via a Cooper-Nathans method in the Reslib library [46]. Uniformlysampled 41× 41× 41 q-points within the ellipsoid wereused for a convolution function in the Reslib library.Finally, the convoluted intensity I(Q,ω) was comparedwith the experimentally obtained Im[χ(Q,ω)] until weget satisfactory convergence of the paramteter.

Throughout the above process, the set ofparameters that best explain the data was determined.In figure 3, 4(d) and 4(e), the overall V-shapesof the spin-dispersions are modelled accurately bycalculations for both compounds. The splitting ofmagnon branches at the zone center are not asnoticeable in the INS data (figure 4(a) and 4(b)).But nevertheless it is fully consistent with theoreticaldispersion curves. The constant-Q cuts in figure2(b)(d) show this consistency more clearly, especiallygiven the tendency for the convoluted I(Q,ω) to haveslightly higher energies due to instrumental resolutionsthan the calculated energies of the two low-lyingmagnon branches. For example, the two measuredpeak positions at Q = (1 0 1) for YFeO3 are at∼1.7 and 2.4 meV, whereas the theoretically calculatedmagnon energies are at ∼1.2 and 2.42 meV. We notethat the calculated energies of magnon branches at themagnetic zone center are consistent with Raman data(∼1.4 and 2.2 meV) [34].

The best fit parameters are given in table 2together with values for TN and the spin cantingangles. In our work, the values obtained for the DMinteractions for YFeO3 are quite different comparedto those of Hahn et. al. [33]. We point out twopossibilities for this discrepancy:

(i) The spin Hamiltonian used in [33] doesn’t includeDM interaction along the c-axis. Since themagnitude of Dab and Dc is similar in RFeO3,they should be considered together.

(ii) The canting angles θ and φ of YFeO3 usedin [33] are much less than the known values(∼0.5◦). Underestimation of the DM vectors istherefore inevitable since they are proportional tothe canting angles (equation 3).

The ratio between DM interaction and exchangeinteraction, D/J , is a criterion that indicates thecompetition between them. A rough estimate for thespin canting angle is given by tan−1(D/J), and so onecan find an approximate value for D/J from equation3. For LaFeO3 the value we obtain for D/J is ∼0.026,which is larger than the values obtained from DFTcalculations (∼0.018 in reference [42], 0.021 in reference[47]). It is also noteworthy that the canting anglesof YFeO3 and LaFeO3 are remarkably similar, which

Low-energy spin dynamics of orthoferrites AFeO3 (A = Y, La, Bi) 7

Table 2. Best fit parameters and spin canting angles used in this work and compared to other work on YFeO3.

TN (K) Jc Jab J′ |Dab| |Dc| Ka Kc θ (◦) φ (◦)

YFeO3 (our work) 644 5.02 4.62 0.22 0.1206 0.1447 -0.0091 -0.0025 0.51 0.58YFeO3 (reference [33]) 644 4.77 4.77 0.21 0.079 - -0.0055 -0.00305 0.30 0.18LaFeO3 (our work) 738 5.47 5.47 0.24 0.130 0.158 -0.0124 -0.0037 0.52 0.46

is quite unexpected because they have significantlydifferent values for their respective FeO6 octahedrarotation angles. In case of YFeO3, the ratio betweencanting angles θ/φ ∼ 1.137 is consistent with previoustheoretical and experimental results [12, 13, 14, 15].

Having said that, the low-energy magneticexcitations of YFeO3 and LaFeO3 have several commonfeatures with that of BiFeO3 (see figure 4(c)(f)and figure 5) such as the shoulder-like signal seenbelow the modes dispersing from the zone center.This feature has been shown to be the result ofcompetition between the three different terms in theHamiltonian: exchange interaction, DM interactionand SIA. Of course, there is room for this featureto manifest itself in several ways depending on thedetails. In Pbnm, centrosymmetricity and local DMvector constrain RFeO3 to have the commensurate 4-sublattice magnetic structure, resulting in the simpleV-shape dispersion curves with two of four magnonbranches as shown in figure 3. In contrast, all local DMinteractions in R3c can be effectively expressed as aglobal DM interaction along two directions, [1 1 0] and[0 0 1]. Thus, a spin cycloid structure can be stabilized.Furthermore, SDW fluctuations and anharmonicityadd more complexity to the structure, making themagnon branches to become more complex. All ofthese effects combined lead to the distinct behavior ofIm[χ(Q,ω)] above 4 meV for RFeO3 and BiFeO3.

Since the magnon branches showing up in the INSsusceptibility of RFeO3 are nearly doubly degenerate:the degeneracy being broken by the DM interaction,there are only two peaks shown in the energy scan. Incontrast, as BiFeO3 has many branches of magnons,the scattering intensity remains high above 4 meV atthe zone center. INS measurements on BiFeO3 withsubstantially improved momentum resolution allowedfor the observation of the individual magnon branches,as shown in the upper part of figure 4(c). The twopeaks at E = 3 meV agree with theoretically calculatedmagnon dispersion, verifying the 4-fold nature of themagnon dispersions.

In YFeO3, LaFeO3 and BiFeO3, some care isnecessary in choosing the proper relative strengths ofthe DM interaction and SIA in order to model the spinwave spectra correctly. The SIA in BiFeO3 is not onlyaffected by the DM interaction, but it is also influentedby various properties such as ferroelectric distortion

and A-site lone-pair effect. This complicated nature ofthe SIA in BiFeO3 has been explained by new DFT-based calculations (see reference [48]). The mixing ofsuch parameters yields a temperature dependence ofseveral properties of BiFeO3, e.g. static propertiessuch as the cycloid periodicity and FE distortion aswell as the dynamical properties such as the spinwave spectrum. Therefore, the spin Hamiltonianparameters of BiFeO3 are also expected to vary asa function of temperature, which has indeed beenobserved [27]. However, both YFeO3 and LaFeO3 donot show any clear temperature dependence of spinwave spectrum (based on our results collected at T =300 K and 1.5 K). This implies that the aforementionedstatic and dynamic properties, and therefore the spin-Hamiltonian parameters of YFeO3 and LaFeO3 remainlargely unchanged between 1.5 and 300 K.

4. Conclusion

The low-energy magnon spectra of YFeO3, LaFeO3and BiFeO3 were studied by our INS experiments.Several features of the magnetic excitation spectrahave been explained by the full spin Hamiltonian,which includes the DM interaction and SIA. Bestfit parameters of spin Hamiltonian were obtained forYFeO3 and LaFeO3. With the careful quantitativeexamination of the magnon behavior in these threecompounds, we have shown how the relationshipsbetween the DM interaction, J , and SIA serves asthe underlying mechanism driving the spin dynamics.Our study provides a guide for future work on otherperovskite systems, in particular with regard to thedelicate balance among DM, J and SIA. The values ofthe magnon mode splitting in most of the other RFeO3compounds is currently available in the literature[29, 32, 33, 34]. Exploiting the relations betweenthese parameters will play a key role in any futureimplementation of technological applications whichutilize Fe3+-based perovskites.

Acknowledgments

We thank Joosung Oh, Y. Noda and D. T. Adroja forfruitful discussions. This work was supported by theInstitute of Basic Science in Korea (Grant No. IBS-R009-G1). The work of M.K. and V.S. was supported

Low-energy spin dynamics of orthoferrites AFeO3 (A = Y, La, Bi) 8

within the program of Large Infrastructures forResearch, Experimental Development and Innovation(project No. LM2015050) and project LTT17019financed by the Ministry of Education, Youth andSports, Czech Republic.

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1 Introduction2 Experimental Details3 Results and Discussion4 Conclusion