Circularly Polarised X-ray Scattering Investigation of Spin-Lattice Coupling in TbMnO 3 in Crossed Electric and Magnetic Fields
H. C. Walker, F. Fabrizi, L. Paolasini, F. de Bergevin, D. Prabhakaran, A. T. Boothroyd, D. F. McMorrow
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Circularly Polarised X-ray Scattering Investigation of Spin-LatticeCoupling in TbMnO in Crossed Electric and Magnetic Fields H. C. Walker,
1, 2, 3
F. Fabrizi,
2, 4, 5
L. Paolasini, F. de Bergevin, D. Prabhakaran, A. T. Boothroyd, and D. F. McMorrow ISIS Facility, Science and Technology Facilities Council,Rutherford Appleton Laboratory, Didcot, Oxfordshire OX11 0QX, UK European Synchrotron Radiation Facility, Boˆıte Postale 220, 38043 Grenoble, France Deutsches Elektronen-Synchrotron DESY, D-22607 Hamburg, Germany London Centre for Nanotechnology, University College London, 17-19 Gordon Street, London WC1H 0AH, UK Diamond Light Source Ltd, Harwell Science and Innovation Campus, Didcot, Oxfordshire, OX11 0DE, UK Department of Physics, Clarendon Laboratory, University of Oxford, UK (Dated: September 10, 2018)We present a study of the magnetic and crystallographic structure of TbMnO in the presenceof crossed electric and magnetic fields using circularly polarised X-ray non-resonant scattering. Acomprehensive account is presented of the scattering theory and data analysis methods used inour earlier studies, and in addition we present new high magnetic field data and its analysis. Wediscuss in detail how polarisation analysis was used to reveal structural information, includingthe arrangement of Tb moments which we proposed for H = 0 T, and how the diffraction datafor H < H C can be used to determine specific magnetostrictively induced atomic displacementswith femto-metre accuracy. The connection between the electric polarisation and magnetostrictivemechanisms is discussed. Similar magnetostrictive displacements have been observed for H > H C as for H < H C . Finally some observations regarding the kinetics and the conservation of domainpopulation at the transition are described. PACS numbers: 75.25.-j, 75.47.Lx, 75.85.+t, 77.80.Dj
I. INTRODUCTION
The magnetoelectric effect, as postulated in 1894 byPierre Curie , in which magnetic and electric order arecoupled together, is obviously a promising candidate forall manner of technological applications . However, forsome forty years after the first experimental observation ,it proved to be little more than a diverting scientific cu-riosity thanks to only a very weak coupling, as a con-sequence of the limited magnetic and electric suscepti-bilities. Therefore, the discovery of a giant magnetoca-pacitance effect in TbMnO in 2003, and the associatedmodern advent in multiferroics, has led to a seismic shiftin the theoretical and experimental research output inthis field, as we have sought to obtain a fundamental un-derstanding of the underlying physics. This is essential ifwe hope to fully exploit the potential for devices, such asmagnetic computer memory which could be written usingan electric field, which would be more energy efficient .TbMnO is not a proper ferroelectric, in which a struc-tural instability towards the polar state, associated withthe electronic pairing, would be the main driving forcebehind the spontaneous polarisation. Instead it is animproper ferroelectric, since the polarisation appears asan accidental by-product of some other ordering, in thiscase magnetic ordering, as implied by the onset of a fi-nite spontaneous ferroelectric polarisation concomitantwith a magnetic phase transition at T = 27 K . Furtherinsight into the driving mechanism was obtained whenneutron diffraction determined that the incommensu-rate magnetic structure goes from a sinusoidal ordering (non-polar) to one described by a cycloid formed by theMn ion spins, which breaks inversion symmetry andmakes TbMnO polar, at the multiferroic phase transi-tion.A similar magnetically driven multiferroic mechanismis present in Ni V O and MnWO , and these ma-terials are collectively classified as cycloidal Type IImultiferroics . The resulting strong magnetoelectriccoupling in TbMnO is demonstrated by the possibilityof controlling the magnetic domain state using an electricfield , and by the switching of the ferroelectric polari-sation axis from ˆ c to ˆ a on application of a magnetic fieldalong the ˆ b axis . Given the clear significance of the mag-netic structure of TbMnO to its magnetoelectric prop-erties, it has been studied in detail using neutrons ,resonant X-ray scattering , and non-resonant X-rayscattering .However, despite all of this progress, until recently the details of the microscopic mechanism driving the fer-roelectricity remained obscured. Two different schools ofthought had developed: one invoked a purely electronicmechanism, where the spin-orbit interaction modifies thehybridisation of the electronic orbitals to generate anelectric polarisation ; whilst the other involves the lat-tice, where the Dzyaloshinskii-Moriya interaction leads toionic displacements, and hence an electric polarisation .Ab-initio density function theory calculations supportedthe significance of the lattice , but any displacementswere beyond the experimental limit of EXAFS ; and itis only through recent developments in non-resonant X-ray scattering that these displacements have finally beenmeasured , providing conclusive support for the pictureof Sergienko and Dagotto .In this paper we provide a comprehensive overview ofthe physics behind the magnetoelectric coupling mech-anism in TbMnO , as obtained through circularly po-larised X-ray non-resonant scattering in crossed electricand magnetic fields. In so doing we present details thatwere omitted from our earlier papers due to spacelimitations, as well as providing new data and analysisto complete our survey of the magnetic field–temperaturephase diagram. The paper is set out as follows: Sec-tion II B explains the experimental technique, giving de-tails of the X-ray polarisation control, and introducesthe non-resonant magnetic scattering amplitude, givinga generic example for a cycloidal spin system, Section IIIreviews the results in zero magnetic field, demonstrat-ing how the technique is complementary to that of neu-tron diffraction for performing complex magnetic struc-ture refinement, Section IV reviews the results in lowapplied magnetic field, and discusses how they may betaken in conjunction with a symmetry analysis to de-termine femtoscale ionic displacements, and finally Sec-tion V presents new data in the high magnetic field po-larisation flop phase. II. NON-RESONANT X-RAY SCATTERINGA. Theory
The interaction between matter and the electric andmagnetic fields of X-ray radiation results in a differentialX-ray scattering cross-section composed of three terms: dσd
Ω = r | F + F NR + F AN | ; (1)where F = P n e i Q · r n f is the charge structure factorwhich contains the Thomson x-rays scattering amplitude f , F NR is the non-resonant (NR) magnetic structurefactor contribution and F AN is the anomalous (AN) orresonant contribution. The radiation interaction withthe magnetic moments is much weaker, such that the in-tensity arising from pure magnetic diffraction is typicallymore than a factor of 10 smaller than the intensity ofclassical Thomson charge scattering. It is for this rea-son, combined with the advantageous element specificity,that most X-ray investigations of magnetic order haveinstead used resonant scattering, exploiting the intensityenhancement observed at the absorption edges, whichcan be from a factor of 100 up to 10 at the rare-earth L -edges and actinide M -edges . However, at the K -edgeof the 3 d transition metals the resonant enhancement istypically only of order four , such that at the absorp-tion edge the scattered intensity will be composed of bothresonant and non-resonant scattering processes, consid-erably reducing the benefits of performing X-ray resonant scattering experiments in this regime, because the non-resonant and resonant x-ray scattering amplitude inter-fere, being of the same order of magnitude. As a result,the interpretation of non-resonant magnetic scatteringis more straight-forward than that of resonant magneticscattering, since it requires no more assumptions thanthe spin and orbital parts of the ordered moment.In this paper we exploit the unique direct coupling ofnon-resonant magnetic X-ray scattering to the magneticstructure, which separates out the spin and orbital partsof the ordered moment, as revealed in the scattering am-plitude for an isolated system of moments : F NR = − i ~ ωmc [ L ( K ) · P L + S ( K ) · P S ] , (2)where L ( K ) and S ( K ) are the Fourier transforms of theatomic orbital magnetisation density and the spin den-sity, and P L and P S are the polarisation factors definedas: P L = − ˆK × [ (cid:0) ˆ ǫ ′∗ × ˆ ǫ (cid:1) × ˆK ] 2 sin θ, (3) P S = ˆ ǫ ′∗ × ˆ ǫ + (cid:16) ˆk ′ × ˆ ǫ ′∗ (cid:17) (cid:16) ˆk ′ · ˆ ǫ (cid:17) − (cid:16) ˆk × ˆ ǫ (cid:17) (cid:16) ˆk · ˆ ǫ ′∗ (cid:17) − (cid:16) ˆk ′ × ˆ ǫ ′∗ (cid:17) × (cid:16) ˆk × ˆ ǫ (cid:17) (4)where k ( k ′ ) is the incident (scattered) wave-vector and ˆ ǫ ( ˆ ǫ ′ ) is the incident (scattered) complex unit vector (inthe case of circularly polarised x-rays), describing thex-rays polarisation state. Simple scattering theory im-plies that the resonant contribution decays slowly, onlyas the inverse of the difference between the resonanceand the photon energy, such that even 5 keV away froma large resonance one might expect a significant resonantscattering contribution. Care should also be taken totake any quadrupolar ( E − E
2) resonant scattering intoaccount . However, when a full analysis of the theoryof magnetic scattering is performed it is revealed thatthe resonant scattering length falls off faster than pre-dicted. For example, for our experiments on TbMnO performed at 6 .
16 keV, the resonant contribution fromthe Tb M IV (E= 1276.9 keV) and M V (E=1241.1 keV)edges is calculated to be only 2 −
5% of the non-resonantcontribution, as explained in the supporting online ma-terial to Ref.[19]. Therefore, by moving away from theedge, it is possible to neglect the resonant term.
B. Experimental Implementation
The experiments were performed at the former ID20magnetic scattering beamline of the European Syn-chrotron Radiation Facility in Grenoble . A single crys-tal of TbMnO , with dimensions 1 × × . (Spacegroup P bnm , lattice parameters at room temperature a =5.3019 ˚A, b =5.8557 ˚A, c =7.4009 ˚A) along the maincrystallographic axes, was synthesized in Oxford usingthe floating zone method. The sample was glued be-tween two copper plates using highly conductive silverpaste to enable the application of an electric field of upto 2 kV/mm along the ˆ c direction. This assemblage wasinserted into either an orange cryostat or the 10T OxfordInstruments cryomagnet giving an a − c horizontal scat-tering plane with the potential to apply a magnetic fieldalong the ˆ b direction. The experiments in zero mag-netic field were performed at an incident X-ray energyof 7 .
45 keV with an Au(222) polarisation analyser crys-tal. An orange cryostat was used in preference to thecryomagnet, so as to have access to a greater region ofreciprocal space, including the complete star of wave-vectors (4, ± τ , ± .The sense of rotation of the circular polarisation statewas defined according to the rule that for θ > θ B thepolarisation rotates in the same direction as the 45 ◦ ro-tation which brings the incident polarisation plane ontothe scattering plane.A complete Poincar´e-Stokes polarimetryanalysis of the scattered beam ( ˆ k ′ ) is ob-tained by collecting the dependence of the in-tensity I ( η ) ∝ P cos(2 η ) + P sin(2 η ) as afunction of η , the rotation angle of a high qualityanalyser crystal about ˆ k ′ . The Stokes parame-ter P = [ I (0 ◦ ) − I (90 ◦ )] / [ I (0 ◦ ) + I (90 ◦ )] is thepolarisation rate parallel and perpendicular tothe scattering plane defined by the analyser, and P = [ I (45 ◦ ) − I ( − ◦ )] / [ I (45 ◦ ) + I ( − ◦ )] the degree ofoblique polarisation . They can be calculated directlyfrom the expression of the complex polarisation vectorsˆ ǫ : P = (ˆ ǫ σ ˆ ǫ ∗ σ − ˆ ǫ π ˆ ǫ ∗ π )(ˆ ǫ σ ˆ ǫ ∗ σ + ˆ ǫ π ˆ ǫ ∗ π ) , P = (ˆ ǫ σ ˆ ǫ ∗ π + ˆ ǫ π ˆ ǫ ∗ σ )(ˆ ǫ σ ˆ ǫ ∗ σ + ˆ ǫ π ˆ ǫ ∗ π ) . (5) C. Calculation of the structure factor for a spincycloid
For example, let we consider a generic incommensu-rate cycloidal spin structure in an orthorhombic cell (seeFig. 1), in which there is a phase shift of π/ S b ˆ b and S c ˆ c which prop-agate according to q = 2 πτ b ∗ = π ˆ b T , where T is themagnetic period. Notice that S contains the K depen-dence of the magnetic spin form factor, and we suppose L =0.Then the incommensurate modulated spin momentmay be written as: S b T=2 π b/q c b d s S c R n r j S FIG. 1. A generic incommensurate spin cycloid (red arrows)in the b − c -plane, propagating along ˆ b directions with peri-odicity T = 2 πb/q , could be described by a magnetic vector ~ S , decomposed in two sinusoidal orthogonal components S b and S c in quadrature, along the crystallographic axis ˆb and ˆc , respectively. S j = S b cos( q · r j )ˆ b + S c cos( q · r j + π/ c = 12 e i q · r j h S b ˆ b + iS c ˆ c i + 12 e − i q · r j h S b ˆ b − iS c ˆ c i = 12 e i q · r j M + 12 e − i q · r j M ∗ (6)where M = ( S b ˆ b + iS c ˆ c ). Inserting this into equation (2),and neglecting the orbital contribution, we obtain: F NR ( K ) = − i ~ ω mc (X n e i ( K + q ) · R n X s e i ( K + q ) · d s M + X n e i ( K − q ) · R n X s e i ( K − q ) · d s M ∗ ) · P S , (7)where the position of the j th atom r j has been decom-posed into R n , defining the position of the n th cell towhich the j th atom belongs, and d s , the position of theatom within the cell ( d s = x s a + y s b + z s c ).By spanning the scattering vector K = k − k ′ onthe basis defined by the reciprocal lattice it is foundthat diffraction occurs for discrete values of K ± q = h a ∗ + k b ∗ + l c ∗ , where h , k , l are integer numbers (Laue’sdiffraction condition), giving rise to twin reflections: X n e i ( K + q ) · R n = 0 → K = h a ∗ + ( k − τ ) b ∗ + l c ∗ X n e i ( K − q ) · R n = 0 → K = h a ∗ + ( k + τ ) b ∗ + l c ∗ , with the magnetic structure factors: F NR ( h, k, l ) ∝ (P s e πi ( hx s + ky s + lz s ) ( M · P S ) P s e πi ( hx s + ky s + lz s ) ( M ∗ · P S ) (8)Clearly the diffraction amplitudes differ for the two re-flections, but whether the measured intensities will differdepends on the polarisation state of the incident beamvia P S . For linear polarised X-rays P S is real and so( M ∗ · P S ) = ( M · P S ) ∗ , with the result that the intensi-ties of the two reflections will be identical. However, forcircularly polarised X-rays P S contains complex polarisa-tion vectors, with the result that ( M ∗ · P S ) = ( M · P S ) ∗ ,and so F NR ( h, k − τ, l )) = F NR ( h, k + τ, l ). Moreover, theleft circular polarisation (LCP) and the right circular po-larisation (RCP) behaviour change symmetrically if thesense of rotation of the cycloid change ( M → M ∗ ) or,equivalently, by considering the opposite magnetic satel-lite − τ . This reveals the power of circularly polarisedX-rays for studying helical, chiral or cycloidal magneticstructures, and allows the determination of magnetic do-main population factor associated to the ratio betweenof clockwise or anti-clockwise cycloids . FIG. 2. a) Circularly polarised x-ray diffraction of a Braggreflection (Thomson scattering). The scattered x-ray intensi-ties have the same η dependence for RCP (green dots) andLCP (red open circles). b) Circularly polarised x-ray mag-netic diffraction of a satellite reflection associated with a mag-netic cycloid propagating in the direction of the scatteringvector K = G + q m . c) Interference between an incommensu-rate displacement wave and the magnetic cycloid at the same K ( q m = q δ ). The ratio between the intensities is t ≈ . Let us consider the simple case described in Fig.2, inwhich we compare the complete Poincar´e-Stokes polari-sation analysis of Thomson and magnetic diffracted in-tensities I ( η ). The incident circular polarisation is de-scribed by the complex vectors ˆ ǫ RCP = (ˆ ǫ σ − i ˆ ǫ π ) and ˆ ǫ LCP = (ˆ ǫ σ + i ˆ ǫ π ). In order to calculate the Poincar´e-Stokes parameters given in Eq.5, we evaluate the finalpolarisation state ǫ ′ , exploiting the Jones matrix for theThomson scattering (see Ref.27): ǫ ′ = (cid:20) θ (cid:21) (cid:20) ± i (cid:21) = (cid:20) ± i cos 2 θ (cid:21) (9)and from Eq. 5 we can calculate the Poincar´e-Stokesterms P i for the scattered polarisations: P = 1 − cos θ θ ; P = 0 . (10)Both the circular polarisations give the same scatteredpolarisation dependence I ( η ), as presented in Fig.2(a),and when the Bragg angle θ = 45 ◦ the scattered lightis completely vertically linear polarised, because P = 1and P = 0. For a different scattering angle, the scatteredpolarisation is elliptical, because a circular contribution P = 0 is present, but still vertical because P = 0.In the case of magnetic cycloid defined in Eq.6, andsupposing that the propagation vector q m = 2 πτ b ∗ par-allel to the scattering vector K , we can calculate theJones matrix from the magnetic scattering amplitude: ǫ ′ = 2 S sin θ (cid:20) − (cos θ − i sin θ )(cos θ + i sin θ ) 0 (cid:21) (cid:20) ± i (cid:21) = 2 S sin θ (cid:20) ∓ ( i cos θ + sin θ )cos θ + i sin θ (cid:21) (11)where we have supposed that S b = S c = S . The Poincar´e-Stokes can be calculated easily from Eq.5 : P = 0; P = ± sin 2 θ. (12)Now the scattered light has a dominant oblique polar-isation P = ± sin 2 θ , with the opposite sign for LCP andRCP, as shown in Fig.2(b). When the scattering angle isclose to θ = 45 ◦ , it becomes completely linear and oblique( P = 1 and P = 0).Finally, it is interesting to analyze the case in whichthere is an interference between the scattering arisingfrom the incommensurate cycloidal magnetic structureand a displacement wave at the same propagation vector q δ = q m , for example as a consequence of the applica-tion of an external magnetic field (see Sec.IV). In thiscase the scattered intensities for RCP and LCP interfereand the Poincar´e-Stokes parameters P i may be calcu-lated by considering the complex sum of the magneticand Thomson Jones matrices. After some easy calcula-tion, the Poincar´e-Stokes parameters may be determined: P = t (1 − cos θ ) ± t (1 − cos 2 θ ) cos θt (1 + cos θ ) ± t (1 + cos 2 θ ) cos θ + 2 P = 2 t (1 + cos 2 θ ) sin θ ± θt (1 + cos θ ) ± t (1 + cos 2 θ ) cos θ + 2 (13)where t is the ratio between the Thomson and magneticscattering amplitude. Fig.2(c) shows the variation of I ( η )for LCP and RCP polarisations for a weak Thomson scat-tering contribution ( t = 0 .
4) superposed to the magneticscattering, revealing the strong sensitivity of this methodto the determination of small displacements associatedto the magnetoelastic coupling, as we will demonstratein Sec.IV. Notice that when t → ∞ we obtain the case ofFig.2(a) valid for the Thomson scattering, whereas when t → III. MAGNETIC STRUCTURE AT H=0 T INTHE INCOMMENSURATE FERROELECTRICPHASE.
In this section, we present the full analysis behindour results previously published in Ref. 11. The zerofield magnetic structure was determined at T = 15 Kin the incommensurate ferroelectric phase, in which theMn atoms develop a cycloidal magnetic structure propa-gating along ˆ b . An essential component of the measure-ments was the poling electric field applied along +ˆ c whilecooling the sample inside the cryostat, enabling us to in-fluence the magnetic domain population. Fig. 3 showsthe final magnetic structure determined by performingcircularly polarised X-ray non-resonant scattering exper-iments (see in Ref.11). An induced weak magnetic con-tribution is also present at the Tb sites, with the sameincommensurate periodicity q m = 2 πτ b ∗ . These resultshave been determined by refining the polarisation depen-dence of scattered intensities for a complete star of wave-vectors (4, ± τ , ± ), a care-ful background subtraction is essential, with the resultthat 36 hours were required to measure the data shownin Fig. 4.Here we shall derive these results and compare themwith the prediction from the model established with un-polarised neutron scattering . We will demonstrate thatthe model does not explain all the observed features andthat a longitudinal magnetic component should be addedto the Tb site. Using a semi quantitative description ofthe structure factor, we shall expand the arguments al-lowing for a description of the magnetic structure. A fitfinally gives the exact parameters.We start by labelling the different magnetic atom po-sitions in the unit cell, as shown in Tab. I.The symmetry splits the Tb moments into two orbits(T1, T6) and (T3, T8), however the moments, followingLandau theory, are taken to be identical. The magneticstructure deduced from neutron diffraction is describedaccording to: FIG. 3. Magnetic structure of TbMnO determined at T =15 K. Only Tb and Mn atoms are shown in the crystallo-graphic P bnm unit cell. The red (blue) arrows refers to theMn (Tb) magnetic moment. (Modified from Ref. 11). !" ! *+,-./--01 ! ! *+,-./--01 + ! " * ! ’1+ " * ! ’1 + ! " *2’1+ " *2’1 / - , * : ; - . / ; - , * : ; - : < * + / = >* ?: FIG. 4. Stokes dependence of the x-ray magnetic scatteringin TbMnO at T = 15 K from the (4 ± τ ±
1) reflections, aftercooling in an electric field along + c , measured with LCP ( • ),RCP (o) and π ( ⋆ ) incident on the sample. The solid linescorrespond to our model calculations, whilst the broken linescorrespond to the model derived by Kenzelmann et al. fromneutron diffraction results (Adapted from Ref.11). m Mn Γ3 = [0 . , . , . µ B m Mn Γ2 = [0 . , . , . µ B m T b Γ3 = [0 , , µ B m T b Γ2 = [1 . , , µ B (14)where the numbers in brackets are error bars, and theabsence of an error bar indicates that a moment com-ponent is forbidden by the cell’s internal symmetry. Theneutron experiment was however insensitive to the phases TABLE I. Mn and Tb atomic positions in the
P bnm unit cell(space group n. 62). ∆
Tba,b describes the fractional atomic co-ordinate along ˆ a and ˆ b of the Wyckoff position (4c) occupiedby the Tb atoms .atom Wychoff a b c labelMn 4 b 1/2 0 0 M60 1/2 0 M11/2 0 1/2 M80 1/2 1/2 M3Tb 4 c 1-∆ Tba ∆ Tbb
Tba
Tbb
Tba
Tbb
Tba
Tbb between several different moment components, includingthe phase difference between the Tb and Mn moments,and that between the b and c components of the Mn mo-ment. With polarised neutrons and X-rays the latterwas shown to be ± π/
2, such that the Mn moments forman elliptical cycloid. Taking the phase of the Mn momentcomponent m b to be zero on site M1, then φ T Ma is thephase of the Tb moments at T1 in the first orbit, whilstthe phase difference φ T Oa between the m a components atT1 and T3 defines the phases of the Tb moments in thesecond orbit.We will now calculate the magnetic scattering ampli-tude based on Kenzelmann’s model for our measuredreflections k ′ − k = (4 , ατ, β ), where α = ± k Miller index, β = ±
1, the sign of l , and thetwo different cycloidal domains are defined by γ = ± γ = +1 corresponds to an anticlockwise rotationwhen moving along +ˆ b and looking from +ˆ a . To con-struct the structure factor for the spins s Mn , s T b , weproceed as for the generic cycloid case given in the pre-vious section adding in the moments on the Tb atomsobtaining: F NR = − i ~ ωmc n s Mnb ˆ b − iαγs Mnc ˆ c )+ βe iαφ TMa sin(8 π ∆ T ba )(1 + e iαφ TOa ) s T ba ˆ a o · P S (15)To facilitate a qualitative comparison of the ex-pected scattering arising from this model structure factorwith the actual non-resonant X-ray magnetic scatteringresults , the above expression can be simplified by as-suming that the scattering angle 2 θ = 90 ◦ , that ˆ b is per-pendicular to the scattering plane and that the other twoaxes of the reference system, i.e. k + k ′ and k ′ − k , corre-spond to the other two crystallographic axes. In writingEq. (15), we skipped all orbital terms since within thissimplified but near real configuration, ˆa · P L is zero, leav-ing for the Terbium, only the spin contribution s T ba . Fur- thermore the Mn orbital moment is taken to be quenchedso that we set m Mn = 2 s Mn . Then the scattering ampli-tudes for a circularly polarised beam incident, measuredin the two linear polarisation channels F σ ′ and F π ′ aregiven by: F σ ′ = M − iǫβ ( υ ′ + iαυ ′′ ) T a F π ′ = iǫM − β ( υ ′ + iαυ ′′ ) T a (16)where ǫ selects the handedness of the incident X-rays,and we use the shorthand notation: M = − i ~ ωmc (2 s Mnb − ǫαγ √ s Mnc ) T a = − i ~ ωmc sin(8 π ∆ T ba ) s T ba / √ υ ′ + iαυ ′′ = e iαφ TMa (1 + e iαφ TOa ) (17)If we consider only the Mn spins then | F σ ′ | = | F π ′ | ,and so P = 0, as in the case described in Eq. 12. Also,as F σ ′ would be real and F π ′ imaginary, this means that | F σ ′ + F π ′ | = | F σ ′ − F π ′ | , such that P = 0. Therefore,any departure from a circularly polarised diffracted beamis interpreted as arising from the Tb moments. We notethat P ∝ | F σ ′ + F π ′ | − | F σ ′ − F π ′ | = − βυ ′ M T a whosesign is left unchanged by reversing the incident polarisa-tion and the sign of τ , while it is reversed by a change in l . Further inspection of the structure factors in Eqs. (16)and (17) reveals that there will be an imbalance in theintensities between the twin reflections ( ± τ ) and for theopposite circular polarisations, where the intensity willbe large for ǫαγ = − ǫαγ = +1, but thehandedness associated with the maximum satellite inten-sity is invariant with respect to changes in the sign of l .We may test these predictions by inspection of Fig. 4,which reveals a clear imbalance in intensities, as expectedfrom the Mn magnetic order. In addition I ( η ) is not in-dependent of η , with extrema near 45 o and 135 o indi-cating that the scattered beam is not wholly circularlypolarised and that P is different from zero. However,the curves from this model do not fit the data so well,and the observed invariance of P when the signs of in-cident polarisation, l and τ are simultaneously reversedcontradicts the model prediction.To reproduce the values of P , the only solution foundwas the inclusion of a Tb magnetic moment componentalong ˆ b , a component to which the earlier neutron diffrac-tion experiments were largely insensitive. In addition wenow take the orbital moments on the Tb ions into con-sideration, with l T b = s T b according to Hund’s rules forthe F electronic configuration.We can now construct the complete scattering ampli-tude: F NR = − i ~ ωmc X s e πi ( hx s + ky s + lz s ) { L s ( K ) · P L + S s ( K ) · P S } = − i ~ ωmc (cid:26)(cid:26) βe iαφ TMa sin(8 π ∆ T ba ) (cid:20) (1 + e iαφ TOa ) 12 l T ba ˆ a (cid:21) + iβe iαφ TMb cos(8 π ∆ T ba ) (cid:20) ( e iαφ T Ob −
1) 12 l T bb ˆ b (cid:21)(cid:27) · P L + nh s Mnb ˆ b − iαγ s Mnc ˆ c i + βe iαφ TMa sin(8 π ∆ T ba ) h (1+ e iαφ TOa ) s T ba ˆ a i + iβe iαφ TMb cos(8 π ∆ T ba ) h ( e iαφ TOb − s T bb ˆ b io · P S o (18) φ T Mb and φ T Ob being defined in the same way as φ T Ma and φ T Oa , the Terbium magnetic structure has fourphase parameters but their freedom may be restricted.The high magnetocrystalline anisotropy of the Terbiumfavours a moment with a fixed direction rather than ro-tating in a cycloid, as confirmed by observations. For ahigh magnetic field applied along b , the Mn momentshave a cycloidal arrangement in the a − b plane, whilethe Tb moments have a fixed oblique direction in thesame plane. Whilst below 7 K, Quezel et al. also foundthat the Tb moments lie along two symmetrical obliquedirections. This implies that φ T Ma and φ T Mb as φ T Oa and φ T Ob differ only by 0 or π , since any other outphasingbetween a and b components would produce a cycloidalrotation. Some arguments developed in Appendix I showthat to obtain the correlation shown in Fig. 4 betweenthe sign of P on one side, and the signs of k , l and theincident polarisation state on the other, we must have φ T Oa = φ T Ob = π , while φ T Mb = 0 , π is preferred. Ourexperiment is insensitive to the a Tb moment compo-nent, as well as to φ T Ma . When the full calculation isperformed, taking the true geometry into account, thebest fit to the data shown by the solid lines in Fig. 4,is obtained for a total Tb magnetic moment componentalong ˆ b of m T bb = 1 . ± . µ B , with a phase shift be-tween the two Tb orbits of φ T Ob = (1 . ± . π and aphase difference φ T Mb = (0 . ± . π between one Mnatom and the subsequent Tb atom moving along ˆ c . Thedomain population is found to be 83(2)% of the cycloidaldomain in which the transverse spiral of the Mn atoms isclockwise, when moving along +ˆ b and looking from +ˆ a .The magnetic structure is not yet completely deter-mined since the phase difference φ T Ma can be 0 or π ,leaving the sign of the Tb a component unknown. In the a − b plane the Tb moment points to a direction near45 o from the axes but in which quadrant remains un-specified. In the ideal cubic perovskite structure, all fourquadrants are equivalent, while in the real arrangementthey are not. Xiang et al predicted from theory an an-gle of 145 degrees between a and the Tb moment labelledT1 in Table 1. IV.
TbMnO < H < H C Neutron diffraction and X-ray resonantscattering measurements have found no evi-dence of a change in the magnetic structure in applied magnetic fields less than the critical field H C for theferroelectric polarisation flop. However, in addition tothe charge reflections observed in X-ray experiments atthe double harmonic positions τ L = 2 τ , which arise froma quadratic magnetoelastic coupling between the spinsand the lattice, in low applied magnetic fields chargereflections were also observed at the single harmonicpositions , which were qualitatively interpreted interms of a linear magnetoelastic coupling. Thereforerather than jumping straight to the H > H C phase,it was decided to investigate the effect of low appliedmagnetic fields first.In this section, we present a complete analysis of ourresults published in Ref. 19. The experiments were per-formed using the same experimental setup as for the zerofield measurements, apart from the fact that the electricstick was inserted into the 10 T Oxford Instruments cry-omagnet rather than an orange cryostat. The samplewas cooled into the multiferroic phase ( T < T N = 28 K)whilst applying an electric field along +ˆ c . Reciprocalspace scans along k over the reflection (4 τ -1) were mea-sured for 6 .
16 keV LCP and RCP X-rays incident, withthe LiF (220) analyser at η = 45 ◦ , 90 ◦ and 135 ◦ , asa function of increasing applied magnetic field (see Fig-ure 5). Below H ≃ η = 90 ◦ are approximately invariant with increasing ap-plied magnetic field, indicating no change in the cycloidaldomain populations. However, the scattered intensity for η = 45 ◦ and η = 135 ◦ is essentially quadratic in the ap-plied field. In another experiment, performed using a Cu(220) analyser at an incident X-ray energy of 6 .
85 keV,we measured the variation in the Stokes dependence as afunction of the applied magnetic field, after poling in anelectric field along − ˆ c (Fig. 6). This demonstrates thatwith increasing the magnetic field the scattering becomesmore linearly polarised.Such a variation cannot be explained by modificationsto the magnetic structure. Instead, this behaviour isdue to the magnetostriction induced ionic displacementsresulting in charge scattering at the magnetic orderingwave-vector . The resultant Thomson scattering ampli-tude varies linearly as a function of field, and the com-bination of the unchanging magnetic scattering and thismagnetic field dependent charge scattering produces theparabolic intensity variation.Indeed let us look at the graph Fig. 5, showing theintensities at η = 45 ◦ and η = 135 ◦ . Their difference ! " $ % &’ ( ) * + ’ , - ) . % - ) * + ’ , - ) , ( ’- / + % ’- ( ! ! FIG. 5. Variation of the scattering from the (4 τ -1) reflectionin TbMnO at T = 12 K as a function of magnetic field forcircular left (top) and circular right (bottom) X-rays incident.Measurements were performed with the analyser at η = 45 ◦ ( ◦ ), 90 ◦ ( △ ) and 135 ◦ ( (cid:3) ). At H = 7 T two peaks are seen,one at the incommensurate position (open symbols) and theother at τ = (full symbols). The intensities for τ = havebeen divided by three to simplify combining the two data setsin the same figure. I n t e g r a t e d I n t e n s it y ( a r b . un it s ) Eta (degrees)5 T4 T3 T2 T LCP5 T4 T3 T2 T RCP
FIG. 6. Stokes dependence of scattering from the reflection(4, τ , −
1) at T = 15 K for 6 .
85 keV LCP and RCP X-raysincident on TbMnO . The lines are fits to equation (19). seems to vary proportionally to the field between -2Tand +2 T at least, and the sign of this variation is oppo-site from one of the incident circular polarization to theother. On Fig. 6 we see that the difference of intensityat η = 0 ◦ and η = 90 ◦ increases with the field, thoughat a different rate for the two opposite circular polariza-tions. Those qualitative features can be simulated usinga scattering amplitude which combines the non-resonantmagnetic scattering with charge scattering: F = ( A C + iB C ) + F NR , (19)where A C + iB C is proportional to the scattering ampli-tude arising from a lattice distortion δ r s : P s exp( i K · [ r s + δ r s ]) f s ( K, ω ).For a qualitative interpretation we may neglect thesmall Tb moment keeping only the Mn and use the samesimplifications as for Eq. (16). Within these assumptions2 θ ≃ ◦ , such that the Thomson term will appear onlyin σ − σ ′ , and we obtain: F σ ′ = M + A C + iB C ,F π ′ = iǫM, (20) M , defined in Eq. (17) is pure imaginary; when changingthe sign ǫ of the incident polarization, it changes its mag-nitude, smaller in circular right, while keeping the samesign. Then | F σ ′ | − | F π ′ | = − iM B C + B C + A C | F σ ′ + F π ′ | − | F σ ′ − F π ′ | = 4 iǫM A C (21)The first and second line explain the features observedFig. 6 and 5 respectively, if assumed that A C and B C are proportional to the field, allowing a simple semi-quantitative determination of their values. TABLE II. Measured real and imaginary part A C and B C , ofthe displacement Thomson structure factor for one cell or 4formula units, in units of r e . The crystal was initially poledin zero magnetic field with an electric field along − c , oppositeto the case discussed in Sect. III. E (keV) h k l A C /H ( r e /T) B C /H ( r e /T) H < .
16 4 τ − . − . τ − . . .
85 4 τ − . − . τ − . . .
77 4 τ − . . τ − . . H = 10 T (Sect. V)6 .
16 4 τ − . − . More accurately we measured the Stokes dependenceof different reflections as a function of magnetic field and,fitting the data with exact formulae, extracted the Thom-son scattering amplitude, shown in Table II. Then by ex-ploiting the variation in the atomic dispersion correctionsfor Mn and Tb at different non-resonant incident ener-gies we obtained a quantitative estimate of the differentoff-centre ionic displacements .Here below, and in Appendix II, we present somearguments that were not fully developed in our previ-ous paper , showing how the magnetostrictive displace-ments are represented in modes with specific symmetryproperties and some phase defined with respect to themagnetic structure. We also discuss the insight whichthe results provide into the origin of the ferroelectric po-larisation.The displacement modes are linear combinations of theatomic displacements, which can be separated into twoclasses, since the magnetic structure splits the eight po-sitions in the P bnm spacegroup (see Appendix II) intotwo independent orbits:∆ α = δ + δ + δ + δ ∆ α = δ + δ + δ + δ ∆ β = δ − δ + δ − δ ∆ β = δ − δ + δ − δ ∆ γ = δ + δ − δ − δ ∆ γ = δ + δ − δ − δ ∆ δ = δ − δ − δ + δ ∆ δ = δ − δ − δ + δ . (22)They may be rewritten according to∆ α ± = 12 (∆ α ± ∆ α ) . (23)In order to describe the symmetry of the modes let usconsider the actions of the generators of the little group G k : 1 2 y m xy b Γ P bnm . How each mode is assigned to one specific irrep,its phase and whether it is visible at the measured peaks(4, ± τ , ±
1) is described in Appendix II, while these prop-erties are summarised in Table III.Since the magnetic structure of TbMnO is describedusing the Γ and Γ irreps, whilst the magnetic field in-duced moment belongs to Γ , the interaction betweenthe two will give rise to two classes of displacements:Γ = Γ ⊗ Γ and Γ = Γ ⊗ Γ , which are shown inFigure 7. The displacement structure factor is even in ± h and ± k , but whilst it is even in ± l for the Γ part,it is odd for the Γ part. Therefore, in order to identifythe particular atomic displacement modes giving rise tothe charge scattering, the Stokes dependence of the scat-tering was measured for two reflections (4 , τ, ± class of modes is complicated since it arises due tointeractions, symmetric and antisymmetric, between nu-merous moments. However, the Γ class arises due to TABLE III. The status of all the displacement modes in bothlow (section IV) and high (section V) magnetic field phases.For each component a , b or c of the modes ∆, the irrep towhich it belong, the phase relative to the magnetic compo-nent M Mnb – R eal or I maginary, and whether they are visibleor extincted for the A-type peak in the experiment are given.The sites column indicates which sites have the visible modein their structure factor. Note: none of the magnetic compo-nents produces any displacement in Γ , but they are includedhere for completeness.mode a b c sites∆ α + Γ ext. Γ R ext. Γ I ext.∆ α − Γ I ext. Γ R ext. Γ I ext.∆ β + Γ R vis. Γ I vis. Γ R vis. O Mn∆ β − Γ vis. Γ I vis. Γ R vis. O O Tb∆ γ + Γ R vis. Γ vis. Γ R vis. O O Tb∆ γ − Γ R vis. Γ I vis. Γ vis. O ∆ δ + Γ I ext. Γ R ext. Γ ext.∆ δ − Γ I ext. Γ ext. Γ I ext. only the symmetric interaction between the induced mo-ment and S T bb , making it more tractable. The differentextracted modes are listed in Table IV, including the dis-placement of the Tb ions along ˆ c , of maximum magnitude − ± TABLE IV. Displacements of the Mn, Tb and oxygen ionsalong the ˆ a , ˆ b and ˆ c axes in femtometres per Tesla in the lowfield phase.Γ iδ Tbb +6 . δ Tba = − ± δ Mnc = − ± δ O c − . δ O a − . iδ O b +3 . δ O a + 0 . iδ O b = − ± δ Tbc = − ± iδ Mnb = +5 ± δ O c − δ O a − . iδ O b − . δ O c = − ± We now get an insight into the onset of ferroelectricityin TbMnO . These Tb displacements are in anti-phasealong ˆ c , and will therefore sum to give zero net ferroelec-tric polarisation, consistent with the plateau seen in themeasured bulk polarisation for H < H C . They resultfrom a symmetric exchange striction between uniform in-duced ∆ m T bb and modulated native m Mnb moments whichare in anti-phase moving along ˆ c . In the zero field struc-ture, m T bb are also in antiphase moving along ˆ c , such thatthe same interaction with m Mnb , gives rise to in-phase dis-placements of the Tb ions along ˆ c , and these contributeto the spontaneous ferroelectric polarisation (see Fig. 4 inRef. 19). Let us suppose that a given force produces thesame displacement, whether magnetostrictive or sponta-neous. Then, by combining our knowledge regarding the0 z y Mn
13 86 Tb z y13 86z y13 86z y1 386 52 47 z y1 386 O2 O1Tb z y1 3 86 (a)
O1O2 O2 z y13 86z y1 386 52 47 z y1 3867 52 4z y1 38 652 47 Tb z y1 3 86 z y Mn
13 86 (b)
O2O2 O2O1 z y Mn
13 86 Tb z y13 86z y13 86z y1 386 52 47 z y1 3867 524z y1 386 52 47 (c) O1 O2O2 O2
FIG. 7. The different displacement modes in (a) Γ , (b) Γ and (c) Γ visible at (4 ± τ ±
1) as listed in Table III. magnetic field induced ionic displacements, the magneti-sation as a function of applied magnetic field , and themagnitude of the Tb moment in zero magnetic field ,we can estimate the zero-field ionic displacements. Aninduced moment equivalent to that of the zero field mo-ment m T bb = 1 µ B occurs for an applied field of 2 . / c to be ofaverage amplitude − ± ± µ Cm − from ionic displacements, representingone quarter of the total polarisation measured . Yet thisshould be taken as just an order of magnitude, becauseassuming that the Tb displacement depends only on theexchange striction force is not quite correct. Indeed acomplete description should include the displacements ofoxygen ions, which behave differently in both cases. Alsowe should remember that this is a secondary effect, occur-ring once the ferroelectric displacement has been triggedby some antisymmetric, such as Dzyaloshinskii-Moriya,interaction. Even though being not precisely quantita-tive, our result brings in a strong argument in favourof a displacive rather than electronic mechanism todrive the ferroelectricity in type II magnetoelectric mul-tiferroics as exemplified by TbMnO . V. TbMnO H > H C Striking evidence for a strong magnetoelectric couplingin TbMnO comes from the observation of a flopping ofthe ferroelectric polarisation from the ˆ c to the ˆ a axison application of sufficiently large ( ∼ FIG. 8. Evolution of the co-existing commensurate and in-commensurate reflections (4 τ -1) in TbMnO as a function oftime in an applied magnetic field H = 7 T at T = 12 K mea-sured with LCP X-rays incident and the analyser at η = 135 ◦ . ! " ! ()*+,-++./ ! " ! ()*+,-++./ - . + * ( : + , - : + * ( : + . ; ( ) - < =( >9 . / )"( ! !=0 ! %/)"(?!=0 ! %/ FIG. 9. Stokes dependence of the scattering from the (4 ± . at T = 14 K and H = 10 T, for LCP( • ) and RCP ( ◦ ) X-rays incident, compared with two models:magnetic structure proposed from neutron diffraction (dot-dash line), and the magnetic structure combined with ionicdisplacements (solid line), where the Thomson component isidentical for the two reflections. fields along the ˆ a or ˆ b axes . Therefore, it is no sur-prise that this transition has been studied in great depthby numerous different probes . Initial scatter-ing studies focussed on the ordering wave-vector, reveal-ing that the polarisation flop transition is concomitantwith a transition from an incommensurate to a commen-surate magnetic structure with k m = (0 , open-ing up the possibility of the ferroelectric polarisationarising due to exchange striction. However, Aliouane et al. , using neutron diffraction, revealed that the flop-ping of the ferroelectric polarisation from the ˆ c to the ˆ a axis in the high field commensurate phase is accompa-nied by the flopping of the Mn spin cycloid from the b − c to the a − b plane . The high-field magneticstructure can therefore be described using the Γ andΓ irreps according to m = (2 . , . , µ B /Mn and m = (0 . , . , µ B /Mn, with a phase difference be-tween the two irreps of 0 . π . The Tb magnetic struc-ture Tb is also modified, from an incommensurate trans-verse and longitudinal sinusoidal wave with componentsalong the ˆ a and ˆ b axes respectively , to an orderingcommensurate with the underlying crystal lattice, i.e. τ Tb = 0, with a total magnetic moment of 7 . µ B com-posed of a 6 . µ B antiferromagnetic component along ˆ a and a 3 . µ B ferromagnetic component along ˆ b . Al-though, it is worth noting that X-ray resonant scatter-ing measurements performed in this commensurate highfield phase revealed scattering at τ = 0 .
25 at the Tb L III edge, implying some ordering of the Tb with this wave-vector . It is therefore likely that the Mn ordering withwave-vector τ = 0 .
25 induces a small moment on the Tbwith the same wave-vector, but the magnitude of this mo-ment is insignificant in comparison with the τ Tb = 0 Tbmoment. Given the strong interest in this phase, it wastherefore fitting to complete our survey of the magneticfield phase diagram of TbMnO using circularly polarisedX-ray non-resonant scattering.If we now return to the previous experiment, Fig-ure 5, then it is clear that the parabolic variation in I ( η = 45 ◦ , ◦ ) in low applied magnetic fields stopsabruptly above 6 T, and then there is a jump in the inten-sity for k = 0 .
25. Whilst at H = 7 T the k -scans revealtwo reflections (Fig. 8), at the incommensurate position k = 0 .
28 and at the commensurate position k = 0 . . On performing the k -scanimmediately after increasing the magnetic field to 7 T, itwas found that the incommensurate reflection was moreintense than the commensurate one. However, when thescan was repeated twenty minutes later, it was found thatthe commensurate peak was now more intense, and thatas a function of time, the incommensurate peak contin-ued to get smaller (see Fig. 8).In another experiment the Stokes dependence was mea-sured at T = 14 K and H = 10 T, in the high field phaseat the (4 ± τ -1) reflections for LCP and RCP incident,see Fig. 9. Before applying the magnetic field, the crystalwas poled in an electric field along − ˆ c . At the end of themeasurement, the magnetic field was lowered back to zerothen the sense of the ferroelectric-cycloidal domain waschecked. In order to obtain a quantitative understand-ing of the measurements shown in Fig. 9, we constructthe amplitude at (4 ± .
25 -1) for the neutron diffractiondetermined magnetic structure , simplified as a Mn el-liptical cycloid with main axes ˆ a and ˆ b ; Tb is not addedsince τ T b = 0: mc − i ~ ω f NR = (cid:16) m Mnb ˆ b − iαγm Mna ˆ a ) (cid:17) · P S . (24)Setting M a = ~ ωmc √ αγs Mna , M b = ~ ωmc s Mnb and goingon with the same simplified geometry as in previous sec-tions, we obtain the scattering amplitudes, including theThomson terms F σ ′ = − iM b + iǫM a + A C + iB C ,F π ′ = ǫ ( M b + ǫM a ) , (25)giving | F σ ′ | − | F π ′ | = A C + B C − M b − ǫM a ) B C − ǫM a M b , | F σ ′ + F π ′ | − | F σ ′ − F π ′ | = 4( ǫM b + M a ) A C . (26)This is to be compared with the data in Fig. 9. The fig-ure includes the simulations made within the exact neu-tron model and the true geometry, with (solid lines) andwithout (dot-dash lines) the Thomson scattering. With-out any Thomson scattering a large P is obtained, whosesign reverts together with the incident polarisation, whilethe experiment shows a large P , keeping the same signin all cases, as predicted by the first of Eqs. (26), if the2Thomson A C , B C terms are dominant. Though smaller,the magnetic terms M a , M b are sufficient to producesome difference between intensities in σ ′ at both incidentpolarisation, through a difference in ( M b − ǫM a ). Fur-thermore the shift of the minimum away from η = 90 ◦ ,revealing a non zero P , is explained by the second ofEqs. (26): since it is known that | M b | > | M a | , we wouldexpect the sign of P to be reversed for LCP vs RCP in-cident, and since M a = 0 the absolute value of P willalso be changed, which is consistent with the asymmet-ric displacement of the minima. These contributions ofmagnetic terms, otherwise known from neutron diffrac-tion, allow finding the sign and scale of Thomson termsthrough a fitting procedure. Moreover, when two fitsare made independently for both (4 ± α , the sign of q . The fit is consistent forone particular sign of the magnetic cycloid only, whichis such determined, see the discussion below. In order todetermine which displacement modes are active withinthe high field phase it is again necessary to perform asymmetry analysis.The parts of the magnetisation with a zero propaga-tion wave-vector are the ferromagnetic b and antiferro-magnetic a components of the Tb moments which bothbelong to irrep Γ . Therefore, since the Mn magneticmodulations belongs to Γ and Γ , the displacementmodes arising through magnetostriction will belong toΓ = Γ ⊗ Γ , as in the phase H < H C , and Γ = Γ ⊗ Γ ,which are listed in Table III and shown in Figure 7.Whilst the same arguments may be used to assign rep-resentations to the modes and determine the extinctionconditions, the relative phase of the magnetic and latticemodulations happens to be less well determined, withthe magnetic structure now locked to the lattice. How-ever, assuming that the locking interaction is a secondorder effect, we still used the same method as for the in-commensurate structure. The phases, given in Table III,are approximate, valid inasmuch as some n glide of thecrystal structure applies to the magnetic structure.Regrettably, with data only for l = − b ,which belongs to the irrep Γ , and hence induces displace-ments in Γ as discussed above. At 6 .
16 keV, the realpart of the Thomson amplitude ( A C from Eqn. (19)) isexpected to come mainly from that common component,since the other displacements contribute to A C via onlya weak atomic dispersion factor. Below H C , at 6 .
16 keV,the fits to the Stokes data give A C /H = − . . µ B /T for the Tb susceptibility, thatequates to an A C of − . µ B on Tb along ˆ b .Meanwhile, in the high field phase Aliouane et al. mea- sured M T bb = 3 . µ B of a total M T b = 7 . µ B at T =8 . H = 5 T . If we extrapolate from this, con-sidering that his total moment is not far from saturation,to our conditions at T = 14 K and H = 10 T, we mayassume a value of M T bb = 4 . ± . µ B at 10 T, whichwould return a value of A C /H = − . ± . A C /H = − . ± . part is concerned, the same magnetostric-tive mechanism applies in the high field as well as in lowfield, i.e. in this phase we might also expect the pres-ence of the Tb displacement along ˆ c along with a similaroxygen displacement arising due to exchange striction .The second interesting observation, obtained from ourfits to the Stokes measurements performed in both theincommensurate and commensurate magnetic phases, isthat the relative populations of the clockwise and an-ticlockwise cycloidal domains are conserved on passingthrough the polarisation flop transition. Going from lowto high field, the plane of the cycloid is rotated by +90degrees about the +ˆ b axis, for a magnetic field along − ˆ b . Then it turns back again when the field is returnedto zero, with the same initial dominant domain restoredwith the same proportion. Such a behaviour was alreadyseen in other experiments . This is somewhat surpris-ing since the symmetry of the system does not favour oneor the other domain of the new phase, in equilibrium withsome domain of the initial phase. And if some dynamicaleffect would be considered at the transition, the domainopposite to the initial one would be preferred when re-turning to zero field. The answer to this conundrum maylie in the insufficiently precise alignment of the magneticfield with the ˆ b axis. Indeed it has been shown that atilt of the magnetic field by 2 degrees in the direction of+ˆ a or − ˆ a could significantly favour one or the other do-main in the phase with ferroelectric polarisation parallelto ˆ a . Such an angle was within the uncertainty of our setup. VI. CONCLUSIONS
In conclusion, we have presented crystallographic andmagnetic structural results for TbMnO , for a magneticfield applied along ˆ b and an electric field along ˆ c , us-ing circularly polarised X-ray non-resonant scattering;bringing together an in-depth analysis of published re-sults with previously unseen data. Starting from a dis-cussion of circularly polarised X-ray non-resonant scat-tering for a generic spin cycloid, we demonstrated in de-tail how polarisation analysis reveals the b − c cycloidin TbMnO in zero magnetic field. This also highlightsthe importance of applying an electric field to producea quasi-mono-domain state. Then for 0 < H < H C weexplained how charge scattering interfering with the non-resonant magnetic scattering allows the determination ofspecific atomic displacements with femto-metre accuracy,revealing the connection between the electric polarisation3and the magnetostrictive mechanisms. Finally we havepresented new data in the polarisation flop phase, whichindicates an interference between charge and magneticscattering from a Mn a − b cycloid, dominated by thecharge. It was nevertheless possible to scale the chargestructure factor with respect to the magnetic one, asin the low field phase, and to determine the cycloidaldomain population.Our analysis shows that the Tb dis-placement along ˆ c is similar in both multiferroic phases,and that the domain populations are preserved on pass-ing through the magnetic field induced phase transition. ACKNOWLEDGMENTS
Thanks to everyone who helped with the experiments,especially A. Fondacaro, J. Herrero-Martin, C. Mazzoli,G. Pepellin, V. Scagnoli, and T. Trenit. We also thank A.Malashevich for enlightening discussions, and T. Kimurafor bringing valuable information to our attention.
APPENDIX I
In this appendix details are given for makinga qualitative comparison between the H = 0 Tdata and the scattering expected for the modelstructure factor, similar to that employed in Sec-tion III’s Eq. (16), while adding in the Terbium b component, with S b =( − i ~ ω/mc ) s T bb cos(8 π ∆ T ba ), L b =( − i ~ ω/mc ) l T bb cos(8 π ∆ T ba ) F σ ′ = M − iǫβ ( υ ′ + iαυ ′′ ) T a + iβ ( ν ′ + iαν ′′ ) S b ,F π ′ = iǫM − β ( υ ′ + iαυ ′′ ) T a − ǫβ ( ν ′ + iαν ′′ )( S b + L b )(27)where and ν ′ + iαν ′′ = e iαφ TMb ( e iαφ TOb − P ∝| F σ ′ + F π ′ | − | F σ ′ − F π ′ | == − βυ ′ M T a − βǫν ′ M L b ++4 α ( − υ ′′ ν ′ + υ ′ ν ′′ ) T a (2 S b + L b ) . (28)The experiment shows that when the signs of k , l andincident polarisation, that is α , β and ǫ , are simulta-neously reversed, P is left qualitatively unchanged (seeFig. 4). This implies that in Eq. (28) the first and thirdterms cancel each other, the second one being non zero.Since the same is true after a sign change of β alone, thefirst and third terms should both be negligible, that is υ ′ ≈ υ ′′ ≈ ν ′ = 0. This in turn sets φ T Oa to π , a valuecompatible with Kenzelmann’s result . φ T Ob differs from φ T Oa by either 0 or π , but only φ T Ob = π gives ν ′ = 0. With these phases our experiment is insensitive to the a magnetic component of Tb and to the phase difference φ T Ma . To maximise | ν ′ | requires that φ T Mb = 0 , π . APPENDIX II
We have used the
P bnm setting for space group ,0,0) x, , ) 0 , , z , (4) 2(0, ,0) , y, , (5) ¯1 0 , ,
0, (6) b , y, z , (7) m x, y, and (8) n ( ,0, ) x, , z .The phase of the displacements is defined relative tothe maximum in the b component of the Mn magnetiza-tion. Consider a glide plane n (operation 8 in the ITCTables for P bnm ) at this maximum, it is a mirror, ei-ther even or odd, both for the magnetic field appliedalong ˆ b , and for the magnetic structure, and hence forthe displacements. The field induced magnetisation ∆ m is invariant under the glide, but all components of themagnetic modulation, m ( x ) at x , will be reversed: n ∆ m = ∆ m , n x = x ′ , n m ( x ) = − m ( x ′ ) , (29)with the consequence for the displacement δ ( x ), propor-tional to m ∆ m n δ ( x ) = − δ ( x ′ ) . (30)For the displacement modes, n exchanges 1 ↔
8, 2 ↔ ↔ ↔
5, and hence ∆ α ↔ ε ∆ α , ∆ β ↔ − ε ∆ β ,∆ γ ↔ − ε ∆ γ and ∆ δ ↔ ε ∆ δ , where ε is +1 for the a , c components and − b . If the sign agrees with that inEqn (30) then the displacement modulation is in phasewith the magnetic one, and is labelled real. Whereasif the signs disagree, this implies that the displacementmodulation is zero rather than a maximum at the maxi-mum in m Mn b , and is labelled as imaginary.The b glide operation exchanges positions: 1 ↔
6, 2 ↔
5, 3 ↔
8, and 4 ↔
7, giving ∆ α ↔ ε ∆ α , ∆ β ↔ ε ∆ β ,∆ γ ↔ − ε ∆ γ and ∆ δ ↔ − ε ∆ δ , where ε is − a component and +1 for b and c ; whilst the action of screwaxis 2 y exchanges differently the mode components. Bycombining these two sets, and using the table for G k , eachcomponent of each mode can be assigned to one specificirrep.Finally, whether a mode is visible at (4, ± τ , ±
1) ornot depends on the action of the n glide. Since the struc-ture factor is calculated for F ( h, , l ) where h + l is odd,only the modes which change their sign under the ac-tion of the glide on the position, irrespective of the valueof the factor ε , are visible, i.e. ∆ β ± and ∆ γ ± . Whenthe geometrical structure factors of the visible modes arecalculated, with the phases found above, they happen tobe purely real for modes in Γ and purely imaginary formodes in Γ and Γ , the latter considered in Section V. P.Curie, J. Phys. 3 , 393 (1894). M. Fiebig, J. Phys. D: Applied Physics , R123 (2005). A. D. N., Sov. Phys.-JETP , 708 (1960). T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima,and Y. Tokura, Nature , 55 (2003). M. Bibes and A. Barth´el´emy, Nat. Mat. , 425 (2008). M. Kenzelmann, A. B. Harris, S. Jonas, C. Broholm,J. Schefer, S. B. Kim, C. L. Zhang, S.-W. Cheong, O. P.Vajk, and J. W. Lynn, Phys. Rev. Lett , 087206 (2005). G. Lawes, A. B. Harris, T. Kimura, N. Rogado, R. J. Cava,A. Aharony, O. Entin-Wohlman, T. Yildirim, M. Kenzel-mann, C. Broholm, and A. P. Ramirez, Phys. Rev. Lett. , 087205 (2005). F. Fabrizi, H. C. Walker, L. Paolasini, F. de Bergevin,T. Fennell, N. Rogado, R. J. Cava, T. Wolf, M. Kenzel-mann, and D. F. McMorrow, Phys. Rev. B , 024434(2010). K. Taniguchi, N. Abe, T. Takenobu, Y. Iwasa, andT. Arima, Phys. Rev. Lett. , 097203 (2006). D. Khomskii, Physics , 20 (2009). F. Fabrizi, H. C. Walker, L. Paolasini, F. de Bergevin,A. T. Boothroyd, D. Prabhakaran, and D. F. McMor-row, Phys. Rev. Lett. , 237205 (2009); , 239902(E)(2011). Y. Yamasaki, H. Sagayama, T. Goto, M. Matsuura, K. Hi-rota, T. Arima, and Y. Tokura, Phys. Rev. Lett. ,147204 (2007); , 219902(E) (2008). N. Aliouane, K. Schmalzl, D. Senff, A. Maljuk, K. Prokeˇs,M. Braden, and D. N. Argyriou, Phys. Rev. Lett. ,207205 (2009). D. Mannix, D. F. McMorrow, R. A. Ewings, A. T.Boothroyd, D. Prabhakaran, Y. Joly, B. Janousova,C. Mazzoli, L. Paolasini, and S. B. Wilkins, Phys. Rev. B , 184420 (2007). J. Strempfer, B. Bohnenbuck, I. Zegkinoglou, N. Aliouane,S. Landsgesell, M. v. Zimmermann, and D. N. Argyriou,Phys. Rev. B , 024429 (2008). T. R. Forrest, S. R. Bland, S. B. Wilkins, H. C. Walker,T. A. W. Beale, P. D. Hatton, D. Prabhakaran, A. T.Boothroyd, D. Mannix, F. Yakhou, and D. F. McMorrow,J. Phys. Condens. Matt. , 422205 (2008). S. B. Wilkins, T. R. Forrest, T. A. W. Beale, S. R. Bland,H. C. Walker, D. Mannix, F. Yakhou, D. Prabhakaran,A. T. Boothroyd, J. P. Hill, P. D. Hatton, and D. F.McMorrow, Phys. Rev. Lett. , 207602 (2009). S. W. Lovesey, V. Scagnoli, M. Garganourakis, S. M. Kooh-payeh, C. Detlefs, and U. Staub, J. Phys. Condens. Matt. , 362202 (2013). H. Walker, F. Fabrizi, L. Paolasini, F. de Bergevin,J. Herrero-Martin, A. T. Boothroyd, D. Prabhakaran, andD. McMorrow, Science , 1273 (2011). H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys. Rev.Lett. , 057205 (2005). I. A. Sergienko and E. Dagotto, Phys. Rev. B , 094434(2006). A. Malashevich and D. Vanderbilt, Phys. Rev. Lett. ,037210 (2008). H. J. Xiang, S.-H. Wei, M.-H. Whangbo, and J. L. F. DaSilva, Phys. Rev. Lett. , 037209 (2008). F. Bridges, C. Downs, T. OBrien, I.-K. Jeong, and T. Kimura, Phys. Rev. B , 092109 (2007). D. Gibbs, D. R. Harshman, E. D. Isaacs, D. B. McWhan,D. Mills, and C. Vettier, Phys. Rev. Lett. , 1241 (1988). E. D. Isaacs, D. B. McWhan, C. Peters, G. E. Ice, D. P.Siddons, J. B. Hastings, C. Vettier, and O. Vogt, Phys.Rev. Lett. , 1671 (1989). L. Paolasini and F. de Bergevin, C. R. Phys. , 550 (2008). M. Blume and D. Gibbs, Phys. Rev. B , 1779 (1988). M. Altarelli, in
Lecture Notes in Physics , Vol. 697(Springer, 2006) p. 201. V. Scagnoli and S. W. Lovesey, Phys. Rev. B , 035111(2009). M. Blume, “Resonant anomalous x-ray scattering theoryand applications,” (Elsevier Science B. V., 1994) p. 495. L. Paolasini, C. Detlefs, C. Mazzoli, S. Wilkins, P. P. Deen,A. Bombardi, N. Kernavanois, F. de Bergevin, F. Yakhou,J. P. Valade, I. Breslavetz, A. Fondacaro, G. Pepellin, andP. Bernard, J. Synch. Rad. , 301 (2007). V. Scagnoli, C. Mazzoli, C. Detlefs, P. Bernard, A. Fon-dacaro, L. Paolasini, F. Fabrizi, and F. de Bergevin, J.Synch. Rad. , 778 (2009). F. de Bergevin and M. Brunel, Acta Cryst.
A37 , 314(1981). C. Mazzoli, S. B. Wilkins, S. Di Matteo, B. Detlefs,C. Detlefs, V. Scagnoli, L. Paolasini, and P. Ghigna, Phys.Rev. B , 195118 (2007). Note that this definition corresponds to P = P ζ , P = P ξ and P = P η in Ref. , and that the electric field of theincident circular ˆ ǫ + c light polarization rotates clockwise foran observer looking towards the source. (), note this is the definition used in magnetic scattering,whilst the convention in crystallography is K = k ′ − k . (), it should be noted that this low count rate is partiallyalso a consequence of the use of the diamond phase plate,which at E = 6 .
16 keV attenuates the incident photon fluxby a factor of ∼ J. Blasco, C. Ritter, J. Garc´ıa, J. M. de Teresa, J. P´erez-Cacho, and M. R. Ibarra, Phys. Rev. B , 5609 (2000). S. Quezel, F. Tcheou, J. Rossat-Mignod, G. Quezel, andE. Roudaut, Physica B+C , 916 (1977). N. Aliouane, D. N. Argyriou, J. Strempfer, I. Zegkinoglou,S. Landsgesell, and M. v. Zimmermann, Phys. Rev. B ,020102(R) (2006). H. Walker, R. Ewings, F. Fabrizi, D. Mannix, C. Mazzoli,S. Wilkins, L. Paolasini, D. Prabhakaran, A. Boothroyd,and D. McMorrow, Physica B: Condensed Matter ,3264 (2009). T. Arima, T. Goto, Y. Yamasaki, S. Miyasaka, K. Ishii,M. Tsubota, T. Inami, Y. Murakami, and Y. Tokura,Phys. Rev. B , 100102 (2005). H. Barath, M. Kim, S. L. Cooper, P. Abbamonte, E. Frad-kin, I. Mahns, M. R¨ubhausen, N. Aliouane, and D. N.Argyriou, Phys. Rev. B , 134407 (2008). T. Kimura, G. Lawes, T. Goto, Y. Tokura, and A. P.Ramirez, Phys. Rev. B , 224425 (2005). N. Abe, K. Taniguchi, S. Ohtani, H. Umetsu, andT. Arima, Phys. Rev. B80