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Symmetry Does Not Allow Canting of Spins in La Sr Mn O 1.4 1.6 2 7 Fan Zhong1,2 and Z. D. Wang1 1Department of Physics, University of Hong Kong, Hong Kong, People’s Republic of China 2Department of Physics, Zhongshan University, Guangzhou 510275, People’s Republic of China∗ (October 28, 1999) We analyze the symmetry of all possible magnetic structures of bilayered manganites 0 La2−2xSr1+2xMn2O7 with doping 0.3 ≤ x < 0.5 and formulate a corresponding Landau theory 0 of thephasetransitionsinvolved. Itisshown thatcantingofspinsisnot allowed at x=0.3though 0 is at x = 0.4. The observed magnetic reflections from the sample with x = 0.3 may be described 2 as arising from two spatially distributed phases with close transition temperatures but different n easy axes and ranges of stability. Experimental results are revisited on the basis of the theoretical a findings. J 4 1 PACS number(s): 75.25.+z, 75.30.-m, 75.40.Cx, 75.30.Kz ] Recent extensive investigationof the so-calledcolossal themagneticstructureseemtoplayanimportantrolein l e magnetoresistance (CMR) [1] in doped perovskite man- the bilayered manganites. r- ganites has stimulated considerable interest in relative For 0.32 <∼ x <∼ 0.4, the bilayered manganites exhibit t s bilayered compound La2−2xSr1+2xMn2O7 in an attempt a FM orderbelowTc with aneasy axisat the layer. The . tounderstandandtoimprovethe sensitivityofthemag- magnetic structure at x = 0.3, however, is somewhat t a netoresistive response [2–4]. The material of interest is complicatedand so there exists no consensus. Perringet m comprised of perovskite (La, Sr)MnO3 bilayers separat- al[10]proposedanAFMorderofanintra-bilayerFMand - ingby(La,Sr)Oblockinglayers,namely,then=2mem- inter-bilayer AFM structure (denoted as AFM-B) with d ber of the Ruddlesden-Popper series of manganites (La, the easy axis along z below about 90K from magnetic n o Sr)O[(La, Sr)MnO3]n. This quasi two-dimensional na- neutron diffraction. However, a substantial component c turepromotesfluctuationsthatlowerthecriticaltemper- withinthelayersrisesupandthenfallsdownbetween60 [ atureT ofthemagnetictransitionandhencetherelevant and90Korso. Argyriouetal[11]byneutrondiffractions c scale of a magnetic field for the huge magnetoresistance. and Heffner et al [12] by muon spin rotation measure- 1 v As the tetragonal I4/mmm symmetry of the material a ments reported, on the other hand, that their sample 3 priori lifts the degeneracyof the eg orbitals of the Mn3+ with the same doping involves two structurally similar 0 ions, the Jahn-Teller distortion of which was argued to phases: The major phase (hole poor) arranges itself in 2 beresponsiblefortheCMRoftheperovskitemanganites a similar AFM-B structure with a substantial canting in 1 [5], observation of antiferromagnetic (AFM) correlations the plane as well as out of it. The minor phase (hole 0 above T of a para- (PM) to ferromagnetic (FM) transi- rich but x < 0.32) differs from the major one only by 0 c 0 tioninLa1.2Sr1.8Mn2O7 wassuggestiveasanalternative its FM arrangement along z axis and its lower ordering / origin to assist localization of carriers above Tc [6]. Im- temperature. However, as they pointed out, the assign- t a portance of the AFM superexchange interaction shows ment of the in-plane component is not so unambiguous. m up at the same doping level as canting of the ordered Also their in-plane AFM reflections become vanishingly moments in neighboring layerswithin eachbilayer as in- small below about 60K either. Still another scenario at - d ferredfromthesignreversaloftheMn-Obondcompress- the 30 percent doping is this: The magnetic structure n ibilitybelowT [7]. Furtherneutronscatteringinvestiga- changes from PM to AFM-B at about 100K and then to c o tion of PM correlations provided evidence for the strong FM at 70K or so. The easy axis rotates correspondingly c canting of the spins with an average angle that depends from in-plane in the AFM-B to z direction in the FM : v on both the magnetic field and the temperature above state [4,13,14]. From these experiments, whether there i X Tc owing to the weaker FM correlation within the bilay- exists canting of spins at x = 0.3 is still ambiguous. So, ers [8]. The canting angle, in particular, changes from noticing the importance of the magnetic structure in the r a 86◦ at zerofield to 74◦ atan externalmagnetic field of 1 x>∼0.4 doping, clarificationof the magnetic structure of ◦ Teslato53 at2Teslasat125K.Comprehensiveneutron- the x = 0.3 doping is a key to understand its character- diffractionstudiesontheotherhandfoundthatthecant- istic transport behavior [3,15]. In this Letter, we show ◦ ◦ ing angle increases from 6.3 at x=0.4 to 180 (A-type that there is a qualitative difference between doping at AFM)atx=0.48at10K,whileT decreasesfrom120K x = 0.3 and x = 0.4 by analyzing the symmetry of the c to 0Kcorrespondingly. Moreover,the AFM correlations magnetic structures. It is found that the symmetry of above T were identified as an intermediate phase whose the magnetic order parameters cannot allow canting at c order parameter decreases in an anomalous exponential x = 0.3 in contrast to x = 0.4. This result sheds new manner upon increasing temperature to about 200K [9]. light to the mechanism of the CMR behavior. Accordingly, the AFM correlations and more generally First we identify the order parameters and their sym- 1 TABLEI. Components of the magnetic vectors that form a basis of theIR’s of I4/mmm at kΓ and kM. c IR BASES τ2 Lz; LAz 2 τ3 Mz; LBz τ9 (Mx,My); (LBx,LBy) τ10 (Lx,Ly); (LAx,LAy) 3 4 1 metry responsible for the possible magnetic structures. b TheMnionswithmagneticmomentsµi intheI4/mmm a structureoccupyfourpositionsati=1(0,0,z),2(0,0,1− z) (z ∼ 0.1) and their translation by t0 = (12,21,12), i.e., (1,1,1±z)(seeFig.1)[16]. Followingtherepresentation FIG. 1. Elementary unit cell of I4/mmm with four Mn 2 2 2 ions and their numbering. analysis of magnetic structures [17–19], we define two magnetic vectors tivistic spin-spin and spin-orbit interactions and so are M=µ +µ effects ofthe orderofO(v2/c2),ordinarilyabout10−2 to 1 2 0 0 L=µ −µ . (1) 10−5,wherev0 isthespeedofelectronsinthecrystaland 1 2 c0 that of light, since the magnetic moments themselves Then a FM state corresponds to M propagating with a wavevectorskΓ =(000),abilayered-typeAFM-Bandan ccoonntsatainntas dfaucetotrovt0h/ecir0 r[e2l0a]t.iviHsteinccoeriαgina.ndbβ aanrde sdmaarlel w A-type AFM (intra-bilayer AFM but inter-bilayer FM) positive for stability. state to M and L, respectively, with k = (001) of the M 2 We now focus on the x = 0.3 doping. The relevant first Brillouin zone. Denoting the latter two order pa- magnetic vectors in this case is L and M. Minimizing rameters as L and L respectively, and noticing that B B A Eq.(2)withthecomponentsofthesevectors,oneobtains kΓ and kM share the same irreducible representations five solutions (IR’s) of the I4/mmm group[21], one can find the components of the four vectors that form bases of the IR’s M=L =0, (3a) B showninTableI. NotethattheIR’sτ9 andτ10 areboth a +α M=0,L =L =0,L2 =− B Bz, (3b) two-dimensional,andsoMx andMy togetherformaba- Bx By Bz b sis vector of τ9, so do LBx and LBy. From Table I and M=0,L =0,L2 +L2 =−aB+αBxy, (3c) the possible experimental magnetic structures [9,11,14], Bz Bx By b wpheaisdee,nMtizfyanLdB(LwBitxh,LthBey)ofrodretrhpeamrainmoertperhafoserotfhxe =ma0j.o3r, LB =0,Mx =My =0,Mz2 =−c+dβz, (3d) (Mx,My) with the order parameter for 0.3 < x <∼ 0.38, c+β ((LMAxx,,MLyA)y)afnodr 0(.L48Ax<∼,LxA<y)0.f5orw0h.i3c8h i<s Ax-ty<pe0A.4F8M, .and LB =0,Mz =0,Mx2+My2 =− d xy. (3e) From Table I, the relevant lowest order magnetic part Since anisotropic terms like M2M2 have not been in- x y of the Landau free-energy can be written as cluded, the direction in the xy-plane cannot yet be de- termined. Note that the exchange term of (L · M)2 c a b d B F = M2+ wL2 + wL4 + M4 type is irrelevant, since M·L = 0 due to the incom- 2 2 w 4 w 4 B Xw Xw patibilityofMandLB alongasingledirection. Eq.(3a) 1 1 represents the PM phase, Eqs. (3b) and (3c) pure AFM- + β M2+ β (M2+M2) 2 z z 2 xy x y B phases with the moments directing respectively along 1 1 the z-axisandthe xy-plane,andEqs.(3d) and(3e)pure + α (L2 +L2 )+ α L2 , (2) (cid:20)2 wxy wx wy 2 wz wz(cid:21) FM phases. An remarkable feature of Eqs. (3) is that Xw there is no mixed order such as L with L or L , Bz Bx By where w represents the summation over L, L , and L . M with M or M and L with M. In other words, A B z x y B Note that the latter two vectors will carrier a factor no canting state exists. The reason is that there is no exp{−ikM ·t0} = −1 when they are translated by t0, symmetry relation between αBz (βz) and αBxy (βxy), so andsocannotappearinoddpowers. InEq.(2), wehave that both L (M ) and L (M ) or L (M ) cannot Bz z Bx x By y separated the exchange contributions (first four terms), simultaneously acquire nonzero values in general. This which depend only on the relative orientation of the can also been seen from Table I that the z and the xy spins, from the magnetic anisotropic energies (remain- components transform according to different IR’s. ing terms), which depend onthe relativedirection ofthe In order to determine the range of stability of the magnetic moments to the lattice andarisefromthe rela- phases, one substitutes the solutions Eqs. (3) into the 2 free energy and obtains respectively to the first order, L2 ≃−aB(λ1−λ4)+bB(α1−α2), (6b) By bB(2λ1−λ3−λ4) FF0 =≃0−, a2B − aBαBz, ((44ba)) L2Bz ≃−aB(λb1B−(2λλ31)−−λb3B−(αλ14−) α2), (6c) Lz 4b 2b B B a2 a α and a similar one in the diagonalplane perpendicular to FLxy ≃−4bB − B2bBxy, (4c) the xy-plane, where we have kept terms of order λ in B B both the numerators and denominators. However, it is c2 cβ F ≃− − z, (4d) readilyseenthatthelefthandsidesofEqs.(6b)and(6c) Mz 4d 2d possessjustoppositesignsingeneral,sothatonlyoneof c2 cβxy them can have a realsolution. Similar result can also be F ≃− − . (4e) Mxy 4d 2d proved by expanding the free energy in the unit vector along L valid at low temperatures. Further, there is B Accordingly, if 0 < β < β , for instance, then F < z xy Mz no external or demagnetizing field to tilt the moments. F andsothemomentswillpointtoz-axis,whereas,if Mxy Therefore, canting is not allowed for the bilayered-type β >β >0,they willlie onthexy-plane. This maybe z xy AFMorderofthemajorphasewithx=0.3doping. The the case for the change of the FM magnetization direc- observation of both the z and the xy components of the tionwithincreasingdopingobservedexperimentally[14]. AFM-Bordershouldthusarisefromthetwophaseseach Similarly,whenα becomesbiggerthanα (bothare Bz Bxy with one kind of the AFM-B components. assumed to be positive without loss of generality), the Nevertheless, mixing of different magnetic vectors is system changes from the phase LBz [Eq. (3b)] to LBxy still possible by coupling ofthe type M2L2 for instance. [Eq. (3c)]. The two phases have respectively crystallo- B This can exist due to either an exchangeor a relativistic graphic space groups P4/mnc and Cmca, which cannot origin. Adding such a term with a coefficient δ/2 for be relatedby an active IR and so the transitionbetween the coupling of, say,M and L and L for the minor z Bx By them is necessarilydiscontinuous[22]. Another reasonis phaseofx=0.3,oneobtains,besidesEqs.(3d)and(3c), that the two directions are not connected continuously. a new phase with mixing In practice, the two phases may appear almost simultaneously within of a single sample at different places δa −cb where there is, for example, a small variation of doping Mz2 ≃ dbB −δ2B, B orinhomogeneitysincethetwophasesdifferintheirtran- δc−da sitionpoints [a +α =0,Eqs.(3)]andfreeenergiesby L2 +L2 ≃ B, (7) only values of Bthe orBder of O(v2/c2), and so which will Bx By dbB −δ2 0 0 appear depend rather sensitively on detailed conditions. where we have neglected α and β. A system with such B This same reason also implies that the separation might a coupling may exhibit several scenarios depending on be mesoscopic. Moreover, the two phases may have dif- the strength of the coupling and the nature of the pure ferent temperature windows of stability due to different phases[23,18]. Itmayappearinapurephase,whichmay variations of α and α with the temperature. Oc- Bz Bxy transform continuously or discontinuously to the mixed currenceofAFM-BorFMorderreliesontheotherhand phase,or discontinuously to another pure phase at lower on whether a or c becomes negative first, respectively. B temperatures,thelattercanonlytakeplaceinthestrong There exists possible mixing of LBz and its xy-plane couplingofδ2 >db . Itmayevenchangedirectly to the B counterpartsathigherorderterms,butitcannotproduce mixedphasewhenthetransitiontemperaturesofthetwo cantingeither. Asthetransitionpointsofthetwophases pure phases get identical. Reentrant phase transitions differ by only small quantities of order of O(v2/c2), we 0 0 from a pure phase to a mixed one and then back to the use the expansion in L itself. Thus, besides those pure B pure phase are also possible. L terms in the free energy Eq. (2), we add terms B We now compare our results with experiments. The experimental assignment of both a canting major phase 1 1 4λ1L4Bz, 4λ2(L2Bx+L2By)2, and a canting minor phase is based on the result that if canting is exclusively associated with only one phase, 1 1 2λ3L2Bz(L2Bx+L2By), 2λ4L2BxL2By, (5) the resultanttotalmagneticmomentis toolargeat80K, near the peak temperature of the plane AFM reflections with the coefficients λ’s oforderO(v4/c4) relative to the [11]. This excludes the possibility of a canting minor 0 0 exchange ones [17]. Then one can obtain new solutions phase and a pure LBz phase and appears to suggest in- that determine the direction of the moments in the xy- steadthattheplaneAFMreflectionsariseatleastpartly plane to be either along thex or y axisor alongits diag- from an independent LBxy phase. The fact that the re- onal depending respectively on whether λ4 is positive or flections from LBz and LBxy start appearing at almost negative. In addition, there appear solutions such as the same temperature seems to support the theoretical resultsthatbothphasesemergealmostsimultaneouslyat L =0, (6a) differentplaceswherethereisasmallvariationofdoping Bx 3 or inhomogeneity, which balances the small quantities creaseinthe L phaseelongatesthein-planescalebut Bxy α and α in their transition temperatures. With shortens the z scale, and then a decrease gives rise to a Bz Bxy the twophasesratherthanasingle cantingmajorphase, reverse effect [11,24]. As both the z and the xy compo- the too large magnetic moment may be remedied. The nentspossessabilayered-typeAFMstructure,the mate- peakstructureofthe reflectionintensities fromtheL rialshouldbe expectedto displayaninsulating behavior Bxy phase [10,11] may then arise from the different temper- in the whole temperature range. So the metal-insulator ature dependence of α and α in such a way that transitionshouldmostlybeattributedtothepercolation Bxy Bz below about 60K, α > α , and so the L phase of the minor FM phase, whose transition temperature, Bxy Bz Bxy transforms to the L phase by a reorientation transi- however,seems to be too low [11]. Further workis desir- Bz tion. Thesmallremainingreflectionsmayoriginatefrom able to clarify this. the remnant L phase due to possible inhomogeneity This work was supported by a URC fund at HKU. Bxy or supercooling. Forhigherdoping,noting thatthe reflectionsfromthe M component emerge separately and accompany with z the decline of the L reflections [11],it seems that the Bxy minor phase may be a pure FM phase with the z-axisas its easy orientation. 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