Neutron Scattering Study of Magnetic Anisotropy in a Tetragonal Antiferromagnet Bi 2 CuO 4
Bo Yuan, Nicholas P. Butch, Guangyong Xu, Barry Winn, J. P. Clancy, Young-June Kim
IIn-plane Magnetic Anisotropy Generated by Quantum Zero-point Fluctuations in aTetragonal Quantum Antiferromagnet Bi CuO Bo Yuan, Nicholas P. Butch, Guangyong Xu, and Young-June Kim Department of Physics, University of Toronto, Toronto, Ontario, Canada, M5S 1A7 NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA (Dated: July 19, 2019)We carried out inelastic neutron scattering measurements to study low energy spin dynamics of atetragonal quantum magnet Bi CuO . Unlike other previously studied cuprates, its unique magneticlattice gives rise to an accidental in-plane spin rotational symmetry, not present in the microscopicHamiltonian. We demonstrate that this accidental symmetry is removed by an in-plane magneticanisotropy produced by quantum zero-point fluctuations using spin-wave theory calculations. Inaddition, we find that the size of the in-plane anisotropy agrees quantitatively with the spin-floptransition field of ∼ CuO . PACS numbers:
Determination of the ground state of a magnetic sys-tem is one of the central goals in the study of mag-netism. In most cases, it is sufficient to treat the problemclassically by minimizing the energy while treating thespins as classical vectors; quantum zero point fluctuation(QZPF) in these cases is responsible for only quantita-tive renormalization of the ground state properties fromtheir classical values [1]. However, QZPF can play a deci-sive role in the determination of ground state in systemswith lower spatial dimensions or in the presence of mag-netic frustration such as quantum spin liquids [2]. Whilemagnetic order is destroyed by QZPF in quantum spinliquids, there exists another scenario where the QZPFactually stabilizes the order. This arises when there isan accidental degeneracy within a continuous manifoldof states on a classical level. This could happen in sys-tems with high crystalline symmetries where anisotropicterms in the microscopic Hamiltonian cancel on average,leaving the mean-field energy functional with a highersymmetry than that of the actual Hamiltonian. Sincethis degeneracy is not protected by symmetry, it can beremoved by an anisotropy generated by QZPF, which se-lects and stabilizes the correct ground state[3–5]. Thiswas recently demonstrated in a three-dimensional frus-trated magnet with pyrochlore structure Er Ti O [6, 7].Despite extensive theoretical efforts, finding clear ev-idence for anisotropy generated by QZPF is challengingbecause small symmetry-allowed perturbations tend tobe present in real materials and compete with the QZPFeffect. Although this phenomena has long been proposedfor layered cuprates with a tetragonal lattice structureand an in-plane magnetic order[8–10], a clear demon-stration of an anisotropy generated by QZPF withinthe easy ( ab ) plane has remained elusive. The pres-ence of an orthorhombic distortion or a second inter-penetrating square lattice provides additional small per- turbations that compete with the QZPF in La CuO [11–14] and Sr Cu O Cl [15–17], respectively. AlthoughSr CuO Cl does not suffer from the above problems, therelative displacement between neighbouring CuO layerslowers the symmetry of the mean-field energy by allowinginter-planar dipolar and pseudo-dipolar interactions[10],which is of the same order of magnitude as the anisotropygenerated by QZPF[15].As shown in Fig. 1, the magnetic Cu ions inBi CuO are arranged in a square lattice stacked di-rectly on top of each other. Structurally, it is similarto undoped YBa Cu O , where QZPF has also beenconsidered[8] but has never been directly observed[18].Its lattice structure remains tetragonal down to thelowest temperature[19]. Below T N ∼
50 K, spins con-fined in the ab plane[20] order ferromagnetically along c and antiferromagnetically in the ab plane as shown inFig. 1[19, 21–23]. By virtue of the four-fold symmetry,the crystal is invariant under a π/ CuO , unlike othertetragonal cuprates such as Sr CuO Cl . The four-fold structural symmetry, which moves both atoms as wellas rotates spins, is therefore equivalent to a four-fold spin − rotational symmetry. The latter guarantees anyin-plane anisotropy to cancel on the mean-field level (SeeSupplementary Materials for more details), thus rulingout the anisotropy terms considered for Sr CuO Cl andSr Cu O Cl as alternative explanations for the observedanisotropy.In this letter, we report our neutron scattering study ofspin dynamics along with spin-wave theory calculationsto show that the in-plane anisotropy in Bi CuO is gener-ated by QZPF. Existence of the in-plane anisotropy is un-ambiguously demonstrated by the observation of a spin- a r X i v : . [ c ond - m a t . s t r- e l ] J u l FIG. 1: (a) Structure of Bi CuO projected onto the yz planeshowing two unit cells along z . z coincides with c while x and y are rotated 45 ◦ with respect to crystallographic a and b . Copper and bismuth ions are shown by blue and purplespheres. The CuO squares are shaded in blue. Oxygen ionsare not shown here (See Supplementary Materials for struc-ture with oxygen ions). Dominant Heisenberg interactions J − J considered in the spin wave calculation are indicatedby dashed lines. (b) Structure of Bi CuO projected onto the ab plane. DM vectors for the exchange path J ( (cid:126)D ) as well assymmetry related bonds are shown with black arrows. Greenarrows indicate directions of ordered Cu moments. flop transition within the easy plane at H c ∼ quantitatively explained with the anisotropy gen-erated by QZPF from our spin wave theory calcula-tion. The spin Hamiltonian used for the calculation in-cludes both the antisymmetric (Dzyaloshinskii-Moriya,or DM) and symmetric anisotropic interactions, andis independently determined by fitting the out-of-planemagnon mode. Our results therefore establish thatBi CuO is an example in which magnetic properties aremodified on a qualitative level by QZPF.Bi CuO single crystal (4.35g) used for neutron scat-tering measurements was grown using the floating zonetechnique. Time-of-flight neutron scattering measure-ments were carried out using the Disk Chopper Spec-trometer (DCS) at the NIST Center for Neutron Re-search (NCNR). Two incident energies of E i =4.9meVand E i =2.3meV were used with an energy resolution of ∼ ∼ E f =5meV. A vertically focussing py-rolitic graphite (PG) monochrometer, flat PG analyzerand a Be filter was used to select incident and final en-ergies. A collimation setting of guide-open-80’-open wasused to achieve an energy resolution of ∼ ac -planein the scattering plane. A 10T vertical field supercon-ducting magnet was used in both measurements to applya field along b axis. The sample temperature was kept at FIG. 2: Neutron intensity near (1,0,0) magnetic Bragg peakas a function of momentum transfers along (1,0,L) (in recip-rocal lattice units) and energy transfer (¯ hω ). Neutrons withincident energy E i =4.9meV were used for (a) and (b), whichwas obtained with applied fields of 0T and 1T along b . (c)was obtained using E i =2.3meV and a field of 1T along b . ∼ E i =4.9meV near the magnetic Bragg peak position(1,0,0) is shown in Fig. 2(a)-(b) as a function of momen-tum along L and energy transfer ¯ hω . A clear disper-sive magnon mode with ∼ b axis, the in-tensity of the ∼ b . The second mode is also highly dis-persive and becomes almost degenerate with the firstmode away from the antiferromagnetic zone center. Incontrast to the gapped mode, this mode is apparentlygapless and its intensity extends down to the incoher-ent background level. The gap cannot be resolved evenin a high resolution measurement using incident neutronenergy E i =2.3meV as shown in Fig. 2(c). The gaplessand gapped magnon mode come from spin fluctuationwithin and out of the ab plane, which is consistent witheasy-plane anisotropy in Bi CuO .To track intensity change of the two modes as a func-tion of applied field, a triple axis measurement was car-ried out at fixed Q =(1,0,0). Neutron intensity as a func-tion of energy transfers are shown in Fig. 3(a) for differ- FIG. 3: (a) Constant- Q scan at Q =(1,0,0) with differentfields (0T-1.2T) along b . Open circle in the main plot is a scanat Q =(1.1,0,0) at zero field, which is used as non-magneticbackground for energy transfer 0 . < ∼ ¯ hω < ∼ Q scan at (1,0,0) from 0.5meV to1.5meV after subtracting the background at (1.1,0,0) withinthe same energy range. Intensity of magnetic Bragg peak andgapless mode have been normalized with respect to the valuesat 0T and 1.2T respectively. ent fields. The peak at ∼ Q scanscorresponds to the gapped out-of-plane magnon modeshown in Fig. 2(a)-(b). Clearly, intensity of this modeis almost independent of applied field. We also plot thescan at Q =(1.1,0,0) obtained at zero field in Fig. 3(a)with open circles. Since the magnon has fairly steepdispersion, at this Q it has dispersed to higher energy(¯ hω > . < ∼ ¯ hω < ∼ Q =(1.1,0,0) data, the scan at Q =(1,0,0) at zero field clearly shows additional inelas-tic intensity below the ∼ Q =(1.1,0,0) was firstsubtracted from the scan at Q =(1,0,0), and then the in-tensity from 0.5meV to 1.5meV was integrated. The in-tegrated intensity is plotted as a function of applied fieldstrength in Fig. 3(b) (solid circle). One can observe thatthe in-plane mode intensity in this energy range almostdoubles from 0T to ∼ Q =(1,0,0) represented by opencircles, which is completely suppressed as the intensity ofthe in-plane mode reaches its maximum. Both the inten-sity of the in-plane mode and the (1,0,0) magnetic Braggpeak intensity changes most rapidly at H c ∼ . T . Itcoincides with the metamagnetic transition observed inbulk magnetization studies, which was attributed to bea spin-flop transition[23]. As we will explain later, the intensity change is entirely consistent with re-orientationof spins due to spin flop transition within the ab plane.If magnetic interactions are truly isotropic within the ab plane, spins can rotate freely and respond to even an in-finitesimal field. Observation of a spin-flop transition ata finite field therefore directly indicates the existence ofa small in-plane anisotropy.To explain these observations, we carry out linear spinwave theory (LSWT) analysis. We have defined unit vec-tors ˆ x , ˆ y and ˆ z in addition to the crystallographic a , b and c axes as shown in Fig. 1(b).Magnetic interactions between S = spins on Cu ions take the form: J (cid:126)S · (cid:126)S + (cid:126)D · ( (cid:126)S × (cid:126)S )+ | (cid:126)D | J [2( ˆ d · (cid:126)S )( ˆ d · (cid:126)S ) − (cid:126)S · (cid:126)S ] , (1)where ˆ d = (cid:126)D | (cid:126)D | . The first term is the usual Heisenbergexchange interaction. They are isotropic and hence donot contribute to magnetic anisotropy. The second andthird terms are the DM and symmetric anisotropic ex-change interactions ( (cid:126)D is called the DM vector). The lasttwo terms introduced by Moriya[25] break full spin rota-tional symmetry and give rise to magnetic anisotropy aswell as a gap in the magnon dispersion. Unlike Bi CuO ,we note that the anisotropic exchange terms in Eq. (1)are not allowed by symmetry in other layered cuprateswhere the super-exchange takes place along a 180 ◦ Cu-O-Cu bond. Much smaller perturbation arising fromCoulomb exchange interaction was included to explainthe magnetic anisotropy in the other cuprates[8]. In ourcalculation, we use the exchange interactions J − J (See Fig. 1) determined by Janson et al [27] J =4.7meV, J =0.85meV, J =0.44meV and J =0.88meV. These val-ues were determined by fitting to full magnon dispersionfrom earlier inelastic neutron scattering[24, 26] and byDFT calculation[27].Since J is the dominant Heisenberg interaction, we ex-pect its anisotropic part (cid:126)D to be the leading anisotropicterm. Anisotropic terms on the other bonds should bemuch smaller and therefore not included in the calcula-tion. Symmetry analyses allow us to set (cid:126)D = ( D , , (cid:126)D = (0 , D ,
0) for bonds directed along ˆ x and ˆ y ,respectively (See Supplementary Materials). Resultingpattern of (cid:126)D vectors are shown as black arrows in Fig. 1.LSWT is carried out as a function of angle φ between theordered moment direction and ˆ x within the xy plane.The out-of-plane magnon gap comes from symmetricansiotropic term in Eq. (1).To reproduce the observedgap size (∆ ⊥ =1.7meV), we require D =0.23 J . On theother hand, the in-plane magnon mode is gapless for all φ values within LSWT.The absence of in-plane magnon gap as well as degen-eracy of all ordering directions within the xy plane isa consequence of an accidental in-plane spin rotational FIG. 4: (a) Ground state energy ( E g ) per structural unitcell as a function of ordered moment direction φ after cor-recting for QZPF. E g has been offset by a constant value-13.522045meV. Solid blue line is a fit to the ground stateenergy using A cos(4 φ ) where A = 0 . µ eV. (b) Two sub-lattice magnetizations (cid:126)M and − (cid:126)M for a domain with orderedmoment parallel to ˆ x at zero field. (c) When a field greaterthan critical field, H c , is applied along b , a spin flop transi-tion happens. The sublattice magnetizations (cid:126)M and − (cid:126)M arere-oriented almost perpendicular to the field direction. α isthe small canting of magnetic moments away from collinearconfiguration towards the field direction. The curved dashedarrows in (b) and (c) indicate directions of in-plane spin fluc-tuations. symmetry of the mean-field free energy. This acciden-tal symmetry is broken by including QZPF. The groundstate energy for each structural unit cell as a function of φ is given by E g ( φ ) = − J + J + J − J ) + 12 (cid:90) F BZ d(cid:126)k (cid:88) i =1 ¯ hω i,(cid:126)k ( φ ) , if we include QZPF. The integral and sum are carriedout over the first Brillouin zone and four magnon modes.The results are shown in Fig. 4(a) for 0 < φ < π/ CuO lattice.In Fig. 4(a), energy of the system is no longer indepen-dent of the ordered moment direction. It is the smallestfor ordered moment along ˆ x (and ˆ y ) and largest for mo-ment along a (and b ). The energy difference is given by∆ E g ≡ E g ( φ = 0) − E g ( φ = π/ ≈ -0.1 µ eV. Becauseof this anisotropy, the in-plane spin fluctuation shouldacquire a small gap. This could not be observed in Fig. 2due to limited energy resolution. However, we will showbelow that this anisotropy explains the spin flop transi-tion observed in Bi CuO .At zero field, ∆ E g pins the spins along ˆ x or ˆ y . InFig. 4(b), we show a magnetic domain with two sub-lattice magnetizations, (cid:126)M and − (cid:126)M parallel to ˆ x (Theanalysis will not change for a domain with ordered mo-ment along ˆ y ). When a magnetic field is applied along b , (cid:126)M and − (cid:126)M are re-oriented almost perpendicular to the field as in Fig. 4(c) if the Zeeman energy gain is suf-ficient to overcome the magnetic anisotropy. This leadsto a spin-flop transition at a finite critical magnetic field, H c . Microscopically, Zeeman energy comes from smallcanting of spins from collinear configuration towards thefield direction (denoted by α in Fig. 4(c)). Classically,we can estimate its magnitude as E Z = − J + J + J − J ) S cos(2 α ) − HS sin( α ) . Minimizing E Z with respect to α gives E Z = − . H in µ eV (H in Tesla). The spin flop transition should occurwhen E Z ∼ ∆ E g , which gives H theoc ∼ ab plane explains theobserved change in (1,0,0) magnetic Bragg peak inten-sity as well as the intensity of in-plane mode shown inFig. 3. At zero field, the ordered moment and in-planemagnetic fluctuation are parallel to ˆ x and ˆ y respectivelyas shown shown in Fig. 4(b). Neutron scattering is onlysensitive to spin component along b , which is 45 ◦ fromˆ x and ˆ y , for scattering near Q =(1,0,0). Therefore, onlyhalf of the Bragg peak and in-plane magnon mode inten-sity at (1,0,0) are detected at zero field. When field along b is greater than H c of spin flop transition, all spins arere-oriented perpendicular to the field. The ordered mo-ments are now almost parallel to a as depicted in Fig. 4(c)and hence do not contribute to Bragg peak intensity at(1,0,0). On the other hand, in-plane spin fluctuation isnow entirely along b . This maximizes the intensity of in-plane mode. Intensity of the in-plane mode for H > H c should be twice the intensity at zero field, in agreementwith the integrated intensity at H=0 and H=1.2T shownin Fig.3. On the other hand, spin fluctuation along c is independent of spin orientations within ab plane, theout-of-plane mode should be unchanged across the spinflop transition. This is also consistent with our resultsin Fig.3. The experimentally obtained value of criticalfield for the spin flop transition H c ∼ H theoc .In conclusion, we have carried out high resolution in-elastic neutron scattering to characterize the low energyspin dynamics in Bi CuO . We note that nature of lowenergy magnetic excitations in Bi CuO have been un-der constant debate. Early inelastic neutron scattering[24, 26, 28] found a large gap in the in-plane magnondispersion. This contradicts the result of antiferromag-netic resonance (AFMR)[29], which found the in-planemode to be gapless. Our results settled this long stand-ing controversy by confirming the AFMR results. More-over, by tracking the intensity of the in-plane magnonmode and magnetic Bragg peak, we directly observed aspin-flop transition at ∼ CuO as a robust exampleexhibiting anisotropy generated by QZPF.Work at the University of Toronto was supported bythe Natural Science and Engineering Research Council(NSERC) of Canada. 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