Magnetic field induced flop of cycloidal spin order in multiferroic TbMnO3: The magnetic structure of the P||a phase
N. Aliouane, K. Schmalzl, D. Senff, A. Maljuk, K. Prokes, M. Braden, D.N. Argyriou
MMagnetic field induced flop of cycloidal spin order in multiferroic TbMnO : Themagnetic structure of the P (cid:107) a phase N. Aliouane,
1, 2
K. Schmalzl, D. Senff, A. Maljuk, K. Prokeˇs, M. Braden, and D. N. Argyriou ∗ Helmholtz-Zentrum Berlin f¨ur Materialen und Energy, Glienicker Str. 100, D-14109 Berlin, Germany Institute For Energy, P.O. Box 40, NO-2027 Kjeller, Norway Institut f¨ur Festk¨orperforschung, Forschungszentrum J¨ulich GmbH, JCNS at ILL, 38042 Grenoble Cedex 9, France II. Physikalisches Institut, Universit¨at zu K¨oln, Z¨ulpicher Str. 77, D-50937 K¨oln, Germany (Dated: December 3, 2018)Using in-field single crystal neutron diffraction we have determined the magnetic structure ofTbMnO in the high field P (cid:107) a phase. We unambiguously establish that the ferroelectric polar-ization arises from a cycloidal Mn spins ordering, with spins rotating in the ab plane. Our resultsdemonstrate directly that the flop of the ferroelectric polarization in TbMnO with applied magneticfield is caused from the flop of the Mn cycloidal plane. PACS numbers: 61.12.Ld, 61.10.-i, 75.30.Kz, 75.47.Lx, 75.80.+q
The antisymmetric Dzyaloshinski-Moriya (DM)interaction[1, 2] between two spins, S i , S i +1 separatedby r i,i +1 , provides for a natural coupling betweenmagnetism and ferroelectricity with the spontaneous fer-roelectric polarization given by P s ∼ r i,i +1 × ( S i × S i +1 )[3, 4, 5]. This mechanism generates ferroelectricityin a wide variety of magnets such as R MnO per-ovskites with R =Gd, Dy, and Tb,[6, 7], spinel chromateCoCr O ,[8] spin-chain cuprate LiCu O ,[9] and hueb-nerite MnWO [10, 11]. In the R MnO manganites theDM interaction results in cycloidal order of Mn-spinsgiving a spontaneous ferroelectric polarization along the c -axis ( P (cid:107) c ) (Fig. 1(a)). The application of magneticfield results in the flop of the polarization from the c - tothe a -axis and highlights a novel control of one ferroicproperty by an other[6]. It has been assumed that thischange in the direction of the polarization reflects theflop of the Mn spin cycloid, implying that the antisym-metric DM interaction continues to be responsible for thepolarization (Fig. 1(b))[3, 12]. However, in this high field P (cid:107) a -phase, the commensurate magnetic wave vectorfor Mn is also compatible with other magneto-electricmechanisms such as exchange striction[13, 14, 15]. Inthis letter we present a determination of the magneticstructure of TbMnO in a high magnetic field in thecommensurate P (cid:107) a phase. We find that the magneticstructure of Mn-spins is characterized by an ab -cycloidthat accurately describes the direction of the observedferroelectric polarization via the DM interaction. Ourfinding validates the model that the polarization flopsfound in the perovskite manganites result from the flopof the Mn-spin cycloid.The manganite TbMnO crystallizes in the orthorhom-bic perovskites structure P bnm . On cooling, below T N =41K Mn spins order incommensurately point alongthe magnetic wave vector τ ∼ . b ∗ [5, 6]. On furthercooling below T S =28K, a c -axis component of the Mnmoment orders with a phase shift of π/ b -component so as to form a cycloidal structure where Mn-spins rotate within the bc plane and around the a -axis as shown in Fig. 1(a)[5]. The axis of spin rotationdefines the DM interaction, S i × S i +1 , while the distance r i,i +1 is parallel to the modulation vector τ . For thistype of spin order, inversion symmetry is broken yieldingfor R =Tb and Dy, a ferroelectric polarization along the c -axis as indeed is observed ( P s (cid:107) c ∼ a × b )[5, 6].It is tempting to assume that the flop in the ferroelec-tric polarization arises from a flop in the Mn-spin spiral,but so far there is no experimental prove for this. Fur-thermore, the field dependence of the Mn-spin spiral isdifficult to analyse as single-ion anisotropy terms of Mnand R need to be taken into consideration as well as the R -Mn interaction. The fact that in this P (cid:107) a phase Mn-spins order commensurately with τ = b ∗ [15, 16] renderthe cycloidal flop model even more complex. For such acommensurate spin structure it has been proposed thatspin frustration and super-exchange induce lattice distor-tions that break inversion symmetry thereby generatingthe observed direction of the polarization at high field[13, 14, 15]. This exchange striction model can be ap-plied only to a commensurate order, ruling out its va-lidity for R =Dy[17]. The observation that infrared elec-tromagnon signals are always polarized along the a − axisirrespective if P s is parallel to the a - or c -axis furtheradds to the debate of the high field magnetic phasesin these manganites[18]. A cycloidal Mn magnetic or-dering that yields P (cid:107) a in zero field has been observedfor Gd . Tb . MnO [19], however, there is no direct ev-idence that the magnetic field flops the Mn-cycloid toyield a P (cid:107) a polarization. It is therefore pressing to estab-lish an accurate model of the high field magnetic struc-ture of these multiferroic manganites.TbMnO single crystals were obtained by re-crystalizing a ceramic rod under Ar atmosphere using anoptical floating-zone furnace. The field and temperaturedependence of the magnetic propagation wave vector wasmeasured on the E4 double axis neutron diffractometerat the BENSC facility of the Helmholtz Zentrum Berlin a r X i v : . [ c ond - m a t . s t r- e l ] F e b FIG. 1: (Color online) (a,b) Illustrations of the two cycloidalMn magnetic structures proposed to cause ferroelectricity inTbMnO . In both cases the magnetic propagation vector τ is parallel to the b − axis. In zero field (panel (a)) cycloidalorder, Mn spins rotate around the a -axis ( S i × S i +1 ) and ro-tate wholly within the bc -plane. The ferroelectric polarizationvia the antisymmetric DM interaction is produced along the c -axis. The high-field magnetic structure determined in thiswork is shown in panel (b), it yields a P (cid:107) a ferroelectric polar-ization that arises from an ab cycloid where Mn spins rotatearound the c − axis. In both panels we also show the magneticordering of Tb-spins. In the zero field case (a), the mag-netic propagation vectors of Tb and Mn-spins are clampedand Tb-spins point along the a -axis forming a SDW [5]. Thecanted antiferromagnetic ordering of Tb-spins for H (cid:107) b =5Tdetermined in this work is shown in panel (b). (c) Here wedepict an anharmonic ab -spiral where the phase difference be-tween spins 1 and 2 is ω = π/
2. Is such a case the anglebetween spins 1 and 2 is different from that between 2 and1+b yielding an alternating scalar product along the b − axis.Amplitudes of Tb and Mn spins are not to scale. using a neutron wave length of λ =2.45˚A and a λ/ b − axis using the HM1 su-perconducting cryo-magnet. Due to the limited view ofthe sample in this magnet, integrated intensities of Bragg FIG. 2: (Color online) Single crystal neutron diffraction mea-surements from the E4 diffractometer. Here we show scansalong (0, k ,1) in reciprocal space as a function of temperaturemeasured in a 5T field applied along the b -axis. The inten-sity of these scans is plotted in color coding with correspond-ing scales above each panel. (a) Portion of the data showingthe temperature dependence of the A-mode reflection (0, (cid:15) ,1).Here the wave number (cid:15) varies from 0.282 at T N to 0.266 at 15K before it locks discontinuously to the commensurate valueof below 11K. The weak third harmonic reflection was alsoobserved in these scans to disappear at this transition. Thesedata are shown in an enhanced color scale on the same paneland in the same location in the T − Q map. (b) Temperatureevolution of the P bnm forbidden reflection (0,0,1). reflections could only be measured within certain regionsof the 0 kl lattice plane, a situation that prohibits accu-rate analysis of the magnetic structure. To overcome thisproblem we conducted measurements using the D23 neu-tron single-crystal diffractometer installed at the InstitutLaue-Langevin (ILL) with λ = 1 . ◦ . For these measurements a single crys-tal of TbMnO was cut into a parallelepiped with dimen-sion 3.0x2.9x3.7mm with each face perpendicular to onecrystallographic direction. The crystal was oriented withthe b -axis parallel to the field. This geometry allowed usto measure reflections with k from -1.8 to 0.25. In to-tal 138 independent nuclear Bragg reflections consistingof 310 individual reflections were collected at H = 0 T and T=8.5K within a range of (0 . (cid:54) sin( θ ) /λ (cid:53) . b -axisto 5T. In these conditions 140 commensurate ( τ = 0)independent Bragg reflections consisting of 160 individ-ual reflections were measured along with 64 independentreflections with τ = b ∗ . Analytical absorption correc-tions for each reflection were made by the Xtal suite ofprogram, while analysis of the magnetic intensities wasperformed with the Fullprof code.In Fig. 2(a) we show measurements of the (0, (cid:15) ,1) re-flection as a function of temperature with H (cid:107) b =5T mea-sured on the E4 diffractometer. The data shows that Mn-spin ordering is first observed at T N =41K with the wavenumber (cid:15) decreasing on cooling. At T c =11K, (cid:15) changesdiscontinuously to yield a commensurate wave vector of τ = b ∗ [15, 16, 20]. Published polarization data showsthat a P (cid:107) c state develops below T S while the polariza-tion flops to P (cid:107) a at T c , coinciding with the transitionto the commensurate magnetic phase[21]. In the samedata we observe that the third harmonic reflection (0,1-3 (cid:15) ,1) rapidly changes its position and disappears also at T c (Fig. 2(a)). On cooling below T N we find an enhance-ment in the intensity of several nuclear reflections andthe appearance of forbidden reflections such as the (0,0,1)shown in Fig. 2(b). As we discuss below these effects arisefrom the ordering of Tb-spins with magnetic propagationvector τ T b = 0.To determine the magnetic structure of TbMnO inthe P (cid:107) a state we utilized the D23 diffractometer. Herethe sample was cooled in zero field to 8.5K and thena H (cid:107) b =5T was applied so as to enter the commensu-rate P (cid:107) a phase that was confirmed by measurementsof the magnetic wave vector. At this field and tem-perature we find that the most intense magnetic reflec-tions with wave number (cid:15) = , have extinction condi-tion h + k =even, l =odd (A-mode)[24], while reflectionswith h + k =odd, l =odd (G-mode) were considerablyweaker. For space group P bnm and wave vector τ (cid:54) b ∗ there are four irreducible representations (irreps) Γ ofthe magnetic symmetry for the Mn-ion [22, 23]. TheA mode reflections are contained only in irreps Γ , Γ and Γ . Analysis of the data using only A-mode reflec-tions and a single irrep did not result in a satisfactoryfit to the data. This lead us to consider combinationsof representations. Of the possible combinations only amodel using Γ ⊗ Γ produced a satisfactory result with R ( F )=7.9% and R w ( F )=8.5%. This coupled irrep hasthe form of ( A x , G y , C z ) ⊗ ( G x , A y , F z ). Our measure-ments showed that F- and C- modes were at the very limitof detection indicating that the Mn moment is essentiallycontained within the ab -plane. Reflections from thesemodes were not included in the final refinements. [25]Our analysis yielded a magnetic structure for Mn givenby the moment in µ B : m Mn = Γ (2 . , . , ⊗ Γ (0 . , . , . π closeto π expected for a cycloid. [26] The values of the Mn-spins in this commensurate cycloidal structure along the a − and b − axis are given in Fig. 3 for the z =0 and z = layers.The magnetic structure indicated by this model isdominated by the A modes of these two irreps ( A x and FIG. 3: (Color online) Results of refinements of the mag-netic structure at 8.5K and H (cid:107) b =5T. In panels (a) and (b)we show the variation of the M x and M y components of theMn-spins for the ions located at z =0 (black) and z = (red)along the magnetic propagation vector. Note that the compo-nents along a and b are out of phase so as to yield a cycloidalordering shown in Fig. 1. In panels (c) an (d) we show thecomparison between observed and calculated magnetic struc-ture factors ( F ) for the analysis of the Mn τ = reflectionsand nuclear plus Tb-magnetic reflections respectively. Sincethe Tb magnetic propagation vector is τ =0 the nuclear andmagnetic reflections are not separated in reciprocal space asin the case for the Mn-ions. A y ) producing an elliptical cycloid, shown in Fig. 1(b).Here the Mn cycloid is contained within the ab -planeand Mn-spins rotate around the c − axis, consistent withthe direction of the ferroelectric polarization along the a − axis ( P ∼ b × c ). The anisotropy of the Mn cycloidin both low field bc and high field ab configurations ap-pears to be very similar. In both cases the A y mode pos-sesses the higher moment (3.9 and 3.79 µ B /M n respec-tively) while the components orthogonal to this mode aresmaller and of the same magnitude (2.8 µ B /M n ). Thisresult shows that the flop of the cycloidal plane does noteffect the anisotropy of the Mn ordering itself. The devi-ation of the phase shift between the two irreps away fromthe ideal value produces an angle between spins of 85 o .Despite this, S i × S i + remains parallel to the c − axis andshould not influence the magnitude of P s significantly.Finally for the Mn ordering we find that the G x and G y modes are active with amplitudes of ∼ µ B /M n and itsnet effect on the over all cycloidal order that produces apolarization along the a − axis is also relatively small.We now turn our attention to the Tb-spin ordering.We find that in the P (cid:107) a phase Tb-spins order com-mensurately with the underlying primitive lattice (i.e. τ T b = 0). Analysis of the commensurate P bnm re-flections clearly showed additional intensity that can bemodelled by the Tb magnetic order. The best fit to themeasured data was obtained for a ferromagnetic align-ment of Tb-spins along the b -axis and antiferromagneticcoupling between nearest-neighbor Tb-spins along the[110] direction (Fig. 1(b)). Analysis of this structureyields a total Tb moment of 7.24(7) µ B , with an anti-ferromagnetic component of 6.07(9) µ B along the a -axisand a ferromagnetic component of 3.92(6) µ B along the b -axis. The ferromagnetic ordering along the b -axis is in-deed evident in magnetization measurements under sim-ilar conditions[21].The work we present here unambiguously proves thatthe commensurate P (cid:107) a phase in TbMnO coincides withan ab Mn spin cycloid for H (cid:107) b =5T. The antisymmet-ric DM interaction in this case does yield a ferroelec-tric polarization along the a − axis as indeed is observed( P s (cid:107) a = τ × ( S i × S i +1 ) = c × b ). We may thus identifythe inverse DM interaction as the main mechanism forthe magnetic-field induced flop of ferroelectric polariza-tion.In a perfect cycloidal magnetic arrangement the ex-change mechanism proposed in references [13, 14] doesnot yield any ferroelectric polarisation, as the scalarproduct ( S i · S i +1 ) of neighboring spins is everywhere thesame. This still holds for the commensurate perfectlycircular spiral. In the case of an elliptical commensu-rate cycloid the exchange mechanism does cause ferro-electric polarization as the scalar product varies alongthe modulation. In our case with a modulation of fourorthorhombic lattice distances the mechanism of [13, 14]may thus yield a finite polarisation along the a -directionwhich, however, should still be small due to the onlyminor deviation from a perfect circular cycloid. The ex-change mechanism may gain further importance in thecase of a very anharmonic cycloid. In the extreme an-harmonic arrangement, where spins point either alongthe b or a directions yielding the sequence along the b-direction shown in fig. 1(c): left, up, right, down, therewill be a very effective exchange striction mechanism asthe scalar product S i · S i +1 , between nearest neighbor-ing Mn ions alternates along the b -axis (in the example offig. 1(c) the alternating angles are 90 and 0 deg. along the b -direction). In the current analysis we have performed,the phase difference between rows of cycloids propagatingalong the b -axis (e.g. in Fig. 1(c), rows starting at ions 1and 2) is fixed by symmetry as ω = πτ / ◦ [23][27].A deviation from this value will yield an anharmonic spi-ral, however such as case can not be fully tested withthe current data and we can not exclude a small degreeof anharmonicity. 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