Partially disordered state and spin-lattice coupling in an S=3/2 triangular lattice antiferromagnet Ag2CrO2
M. Matsuda, H. Yoshida, M. Isobe, C. de la Cruz, R. S. Fishman
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Partially disordered state and spin-lattice coupling in an S =3/2 triangular latticeantiferromagnet Ag CrO M. Matsuda and C. de la Cruz
Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
H. Yoshida and M. Isobe
National Institute for Materials Science (NIMS), 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan
R. S. Fishman
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA (Dated: November 9, 2018)Ag CrO is an S =3/2 frustrated triangular lattice antiferromagnet without orbital degree offreedom. With decreasing temperature, a 4-sublatice spin state develops. However, a long-rangepartially disordered state with 5 sublattices abruptly appears at T N =24 K, accompanied by a struc-tural distortion, and persists at least down to 2 K. The spin-lattice coupling stabilizes the anomalousstate, which is expected to appear only in limited ranges of further-neighbor interactions and tem-perature. It was found that the spin-lattice coupling is a common feature in triangular latticeantiferromagnets with multiple-sublattice spin states, since the triangular lattice is elastic. PACS numbers: 75.25.-j, 75.30.Kz, 75.50.Ee
I. INTRODUCTION
Geometrically frustrated magnets give rise to many in-teresting phenomena originating from the macroscopicground state degeneracy. Spin liquid ground state isexpected in strongly frustrated quantum kagom´e andpyrochlore lattice Heisenberg antiferromagnets.
Al-though quantum triangular lattice Heisenberg antiferro-magnet shows 120 ◦ long-range magnetic order with 3 sub-lattices, additional multiple-spin interactions can leadto spin liquid state. Triangular lattice Heisenberg an-tiferromagnets with classical spins also show interestingphenomena related to the chirality degree of freedom. Amagnetic transition driven by the binding-unbinding ofthe Z vortices is predicted to occur at a temperature,where the vortex correlation length diverges but the spincorrelation length remains finite. Spin-lattice coupling sometimes plays an importantrole in selecting the ground state in the frustrated mag-nets. Even if there is no orbital degree of freedom, asmall amount of structural distortion is sufficient to liftthe ground state degeneracy and stabilize a long-rangemagnetic order. For example, in the Cr-based spinels,consisting of the pyrochlore lattice, the magnetic or-der is observed far below the Curie-Weiss temperatureΘ CW due to spin-lattice coupling. Furthermore, a half-magnetization plateau is stabilized in a wide range ofmagnetic field by the spin-lattice coupling.
Similareffect is also predicted for the triangular and kagom´e lat-tice antiferromagnets. Ag M O ( M =Cr, Mn, and Ni) consists of triangu-lar lattice planes of M O , which are well separatedby the metallic Ag layers. Ag NiO (Ni , d low spin S =1/2) and Ag MnO (Mn , d high spin S =2), which have orbital degree of freedom, were stud- (a) (b)(c) FIG. 1. (Color online) (a) Crystal structure of Ag CrO intrigonal ( P ¯3 m
1) phase above T N (=24 K). 2 × × triangular lat-tice plane in a single unit cell. (b) and (c) show schematicstructures of the triangular lattice plane of the Cr spins intrigonal ( T > T N ) and monoclinic phases ( T < T N ), respec-tively. Magnetic interactions ( J , J , and J ) are shown in(b). ied previously. Both compounds show Jahn-Tellerdistortions, which affect the magnetic ground state. InAg NiO , a long-range magnetic order with Q m =( , ,0) hexagonal occurs below N´eel temperature T N =56 K. InAg MnO , a glassy state with a short-range magnetic or-der with Q m =( , , 0) monoclinic is observed as the groundstate.Ag CrO has a trigonal structure ( P ¯3 m
1) with a =2.9298 ˚A and c =8.6637 ˚A at T =200 K, as shown inFig. 1(a), and consists of regular triangular lattices ofthe magnetic Cr (nominally t g ; S = 3 /
2) ions with-out orbital degree of freedom. Therefore, Ag CrO is agood candidate for a model compound of a regular tri-angular lattice Heisenberg antiferromagnet. The specificheat shows a sharp peak at T =24 K. The magneticsusceptibility shows that Θ CW = −
97 K and the effec-tive moment p eff = 3.55 µ B , consistent with 3.87 µ B ex-pected for S =3/2. These results indicate that an an-tiferromagnetic long-range order develops below T N =24K. In addition, the magnetic susceptibility shows weakferromagnetic component below T N .We performed neutron diffraction experiments on apowder sample of Ag CrO . It was found that Ag CrO shows a partially disordered (PD) state with 5 sublatticesand a structural distortion simultaneously below T N =24K, indicating a spin-lattice coupling to stabilize the N´eelorder. We also measured the magnetic diffuse scatter-ing and found that the material does not prefer the 5-sublattice (5SL) structure but 4-sublattice (4SL) struc-ture above T N . The structural distortion is consideredto modify further-neighbor interactions so that the 5SLstructure becomes stable. As far as we know, Ag CrO is the first quasi-two-dimensional (2D) triangular latticeantiferromagnet that shows the long-range PD state. II. EXPERIMENTAL DETAILS
A powder sample of Ag CrO was prepared by thesolid state reaction technique with stoichiometric mix-ture of Ag, Ag O, and Cr O powder in high pressure. The powder sample that weighs ∼ λ ) 1.5374 and 2.410 ˚A for struc-tural analysis and high-resolution measurement, respec-tively. Magnetic diffuse scattering was measured on athermal triple-axis neutron spectrometer HB-1, installedat HFIR in ORNL. The horizontal collimator sequencewas 48’-80’-S-80’-120’. The fixed incident neutron en-ergy was 13.5 meV with the energy resolution (∆ E ) of1.2 meV. Contamination from higher-order beams waseffectively eliminated using PG filters. III. EXPERIMENTAL RESULTS
Figure 2 shows the neutron powder diffraction patternsin Ag CrO at 4 and 45 K, measured with a high resolu-tion mode ( λ =2.410 ˚A). The magnetic Bragg reflections,observed below T N , are shown in Fig. 2(a). The magneticreflections with
15 15 L and
45 15 L , where L =0 and 1, wereobserved, indicating that the magnetic structure has a5SL state in the triangular plane and also the unit cellalong the c axis is the same as the chemical one so thatthe magnetic arrangement is ferromagnetic along the c I n t en s i t y ( a r b . un i t s ) ! (degrees) T (b) M M I n t en s i t y ( a r b . un i t s ) T / / m / / m T / / m / / m T (a) I n t en s i t y ( c oun t s / m i n . ) I n t en s i t y ( c oun t s / m i n . ) T (K) T (111.7 deg.)1/5 1/5 0 m (19.8 deg.) FIG. 2. (Color online) Neutron powder diffraction patternsin Ag CrO at 4 and 45 K measured with λ =2.410 ˚A. Thedata at low scattering angles show that some magnetic Braggpeaks develop below T N (a) and those at high scattering anglesshow that a nuclear Bragg peak splits below T N (b). The insetshows the temperature dependence of the magnetic intensityat 19.8 ◦ and nuclear intensity at 117.9 ◦ , measured with in-creasing temperature. No hysteresis was observed. “T” and“M” denote that the indices are for nuclear reflections in trig-onal and monoclinic phases, respectively. “m” denotes thatthe indices are for magnetic reflections. The magnetic reflec-tions are indexed on the basis of the trigonal structure. axis. Figure 2(b) shows that a nuclear Bragg reflection110 splits into two below T N . The inset in Fig. 2(b)shows the temperature dependence of
15 15 T N , whereas the nuclearintensity drops, indicating that the magnetic and struc-tural phase transitions occurs simultaneously at T N . Thisresult indicates that the spin-lattice coupling releases thehighly frustrated state and gives rise to a N´eel state.Figure 3 shows the neutron powder diffraction patternat T =4 K in Ag CrO , measured with λ =1.5374 ˚A. Weperformed a Rietveld refinement, using the Fullprof pack-age. In the refinement, monoclinic structure with C /m symmetry, which is a subgroup of P ¯3 m
1, is assumed. Theschematic structures for the high-temperature trigonaland low-temperature monoclinic phases are shown in Fig.1(b) and 1(c), respectively. In the monoclinic phase, thetriangular lattice is slightly contracted along the b axis.The structural parameters determined by the refinement Obs.Calc.DifferenceNuclearMagnetic I n t en s i t y ( a r b . un i t s ) ! (degrees)T=4 K FIG. 3. (Color online) Rietveld refinement of the neutronpowder diffraction pattern measured with λ =1.5374 ˚A at T =4 K, where nuclear and magnetic Bragg peaks coexist.TABLE I. Structural parameters of Ag CrO determined bythe Rietveld refinements at T =4 and 45 K. T =45 K, Trigonal ( P ¯3 m x y z U (˚A )Ag 2 d a d a =2.9246140(7) ˚A, b =2.9246140(7) ˚A, c =8.6610155(1) ˚A R nuc =6.80% T =4 K, Monoclinic ( C m )position x y z U (˚A )Ag 4 i a i a =5.07976(16) ˚A, b =2.916985(9) ˚A, c =8.66164(2) ˚A, β =90.0723(3) ◦ M Cr =2.9(1) µ B R nuc =6.66%, R mag =11.5% are shown in Table I. For the magnetic structure, a PDmagnetic structure with 5 sublattices, shown in Fig. 4, isassumed. In this model spin arrangement is up-down-up-down-disordered-..... along the b axis. The spins are as-sumed to point along the c axis with a ferromagnetic spinarrangement along this axis. The weak ferromagneticcomponent observed in the magnetization measurementis consistent with the PD model since it probably origi-nates from the small ferromagnetic moments induced atthe disordered sites in magnetic field. Both the nuclearand magnetic Bragg reflections are fitted with the mod-els reasonably well. The Bragg R -factors for crystal andmagnetic structures are R nuc =6.66% and R mag =11.5%,respectively. The ordered magnetic moment at the or-dered spin sites was fitted to be 2.9(1) µ B , which almostcorresponds to the full moment for S =3/2.The structural distortion, which is accompanied by themagnetic transition, is unexpected because the Cr ionshave no orbital degree of freedom. This behavior is very up down disordered ba FIG. 4. Schematic spin structure of the PD state with 5sublattices below T N . The ordered spins point along the c axis almost perpendicular to the triangular plane. The dashedlines represent the magnetic unit cell. similar to that observed in the highly-frustrated Cr-basedspinel A Cr O ( A =Zn, Cd, and Hg), in which the spin-lattice coupling stabilizes the magnetic structure both inambient and high magnetic field. It is also reported that a triangular lattice antiferro-magnet CuFeO , in which Fe ion with S =5/2 has noorbital degree of freedom, shows a very similar behav-ior. In CuFeO , a monoclinic lattice distortion elon-gates the triangular lattice along the b axis and the spinstructure is a 4SL state (up-down-up-down-..... alongthe b axis). Since the antiferromagnetic coupling be-tween nearest-neighbor (NN) Fe spins is mediated bythe Fe-O-Fe superexchange interaction, the lattice elon-gation along the b axis increases the bond angle of theFe-O-Fe coupling so that the antiferromagnetic couplingalong the b axis is enhanced. As a result, the 4SL spinstructure is stabilized. In Ag CrO , one side of the tri-angle along the b axis is contracted by 0.4% at T =4 K,as shown in Fig. 1(c). The antiferromagnetic couplingbetween NN Cr spins is mediated by direct Cr-Cr ex-change interaction. Therefore, the contraction along the b axis enhances the antiferromagnetic coupling along the b axis. As shown in Fig. 4, the 5SL structure can bestabilized by enhancing the antiferromagnetic couplingalong the b axis. It is very interesting that the triangularlattice is elastic and can expand or contract to stabilizelong-range magnetic order. The spin-lattice coupling is acommon feature in triangular lattice antiferromagnets inwhich further-neighbor interactions give rise to multiple-sublattice states. IV. ORIGIN OF THE PARTIALLYDISORDERED PHASE
What is the origin of the PD state with 5SL inAg CrO ? Further-neighbor interactions should be rel-evant to such an unusual magnetic state. Mekata etal. performed Monte Carlo calculations in the triangu-lar lattice Ising magnet with J , J , and J , shown inFig. 1(b). The 5SL state is not stable at T =0 Kbut can appear at finite temperature when J and J areantiferromagnetic and relatively large, although the re-gion where the phase exists is very narrow. The 4- or8-sublattice state becomes more stable at low tempera-tures. It was also reported that the Heisenberg modelfor nearest-neighbor spins including a spin-lattice cou-pling explains the collinear 4- and 8-sublattice states. As a result of the spin-lattice coupling, the effectivespin Hamiltonian becomes an Ising model with nearest-,second-, and third-nearest neighbor coupling, J (1 − c ), cJ , and cJ , respectively, where c is a spin-lattice cou-pling. In CuFeO large J ( ∼ J ) and J ( ∼ J )were actually observed. This is because the overlap be-tween Fe and O orbitals is relatively large. However,in Ag CrO , in which e g orbitals of the Cr ion arenot occupied, the overlap of the Cr and O orbitals issmall. Therefore, J and J , corresponding to Cr-O-O-Cr super-super exchange interactions, are not expectedto be so large compared to those in CuFeO . It canbe deduced that the non-negligible J and J originatefrom the RKKY interaction mediated by the metallic Ag layers. The interlayer couplings, which give rise to thelong-range magnetic order, is also considered to originatefrom the RKKY interaction. An abrupt drop of the resis-tivity around T N17 indicates a strong coupling betweenthe magnetic CrO and metallic Ag layers. Theoret-ical studies are desirable to clarify the interesting cou-pling between the two layers. Further inelastic neutronscattering study is also needed to determine the further-neighbor interactions.The PD state has been reported in the Ising spinchains, which form the triangular lattice, such asCsCoBr , CsCoCl , and RbCoBr which are an-tiferromagnetic chain compounds and Ca CoRhO thatis a ferromagnetic chain compound. These compoundsshow PD state with 3 sublattices. In particular,Ca CoRhO shows a PD spin structure, in which thespin arrangement along the c axis is ferromagnetic, asin Ag CrO . In Ca CoRhO , the PD state appears at T =90 K and the disordered spins randomly freeze at T =30 K. The magnetic susceptibility shows a weak fer-romagnetic component with a large hysteresis betweenmeasurements with field-cool (FC) and zero-field-cool(ZFC) processes below ∼ T . On the other hand, themagnetic susceptibility in Ag CrO shows a weak fer-romagnetic component with a tiny hysteresis below T N down to 2 K, indicating that the PD state persists at 2K. It is probable that the disordered spins fluctuate evenat low temperatures due to strong frustration and hightwo-dimensionality. As far as we know, this is the firstobservation of the long-range PD state in quasi-2D trian-gular lattice antiferromagnet. The spin-lattice couplingis considered to stabilize the PD state with 5SL, whichis expected to appear only at finite temperatures in alimited range of J and J and has never been observedexperimentally. -400-2000200400600800 0 0.5 1 1.5 2I(26 K)-I(100 K)4SL (2D, ! =5 Å)3SL5SL I n t en s i t y ( c oun t s / s e c . ) Q (Å -1 ) / / m / / m / / m FIG. 5. (Color online) Magnetic diffuse scattering at T =26 Kabove T N . The scattering at T =100 K was subtracted. Thenegative intensity at low Q probably originates from the para-magnetic scattering that increases at low Q at high temper-atures. The dotted, solid, and broken curves are the resultsof model calculations for 3-, 4-, and 5-sublattice states with a2D spin correlation length ξ =5 ˚A, respectively. We found the interesting spin-lattice coupling that sta-bilizes the PD state with 5 sublattices. As the next step,it is interesting to clarify whether the 5SL state is pre-ferred above T N . We studied how the spin correlationdevelops with decreasing temperature. Figure 5 shows aneutron diffuse scattering spectrum, observed at T =26K, which is 2 K above T N . The spectrum measured at T =100 K is subtracted as a background. A broad peakcentered around 1.2 ˚A − is observed. The broad peakwas simulated using the powder averaged 2D squaredLorentzian. Magnetic form factor was also included inthe calculation. Since the peak width is much largerthan the instrumental resolution ( ∼ − ), the res-olution correction was not performed. The solid curve,which is the result of a model calculation for 4SL statewith a 2D spin correlation length ξ =5 ˚A, reproduces theobserved data reasonably well. Considering the param-agnetic contribution, which is oversubtracted, the agree-ment would become improved. The model calculationswith 3-sublattice (3SL) and 5SL states do not describethe observed data. This result suggests that the spinsystem prefers 4SL state just above T N . Therefore, thefurther-neighbor interactions are considered to be alreadydominant above the structural transition temperature.In the J - J - J model, the energies per spin for the4SL and 5SL phases are evaluated to be E = J − J + J − J z + J ′ z and E = J + J + J − J z + J ′ z ,respectively, where J is negative for antiferromagneticinteraction. Here, J z and J ′ z represent nearest- andsecond-nearest-neighbor couplings between the triangu-lar lattice layers, respectively. Our results indicatethat E < E above T N . Therefore, the relation of4 J − J − J z + 4 J ′ z < J , occursat T N . We discuss the magnetic structure using averaged J ’s, since there is no theoretical calculation for the J - J - J model in the distorted lattice. From the formulas of E and E , J , J , and J ′ z are considered to be influen-tial to select one from 4SL and 5SL states. As mentionedabove, J enhanced along the b axis stabilizes both 4SLand 5SL states. In order to stabilize the 5SL state morethan the 4SL state, | J / J | should increase. Since J perhaps originates from the RKKY interaction, theabrupt decrease in resistivity around T N17 is consideredto change the RKKY interaction and enhance J . Anincrease of J ′ z also stabilizes the 5SL state. It is notedthat the PD state is stable only at finite temperatures.Other collinear states such as 8-sublattice phase shouldbecome stable at T =0 K. Although a PD statewith 9-sublattices is predicted between the 4SL and 5SLphases, it was not observed experimentally.
V. CONCLUDING REMARKS
We have studied spin correlations and spin-lattice cou-pling in a triangular lattice antiferromagnet Ag CrO . With decreasing temperature, a short-ranged 4SL spinstate first develops. At T N , spin-lattice coupling givesrise to a long-range PD state with 5 sublattices, whichis expected to appear only in limited ranges of J , J ,and temperature. Since this is the first observation ofthe long-range PD phase in quasi-2D triangular latticemagnet, theoretical progress in this field is highly desir-able. ACKNOWLEDGMENTS
We would like to thank Profs. Y. Maeno and Y. Mo-tome for stimulating discussions. The work at ORNLwas sponsored by the Scientific User Facilities Divisionand Materials Sciences and Engineering Division, Officeof Basic Energy Sciences, U. S. Department of Energy. See, for example,
Frustrated Spin Systems , edited by H. T.Diep (World Scientific, Singapore, 2004). R. Moessner and J. T. Chalker, Phys. Rev. Lett. , 2929(1998); Phys. Rev. B , 12049 (1998). B. Canals and C. Lacroix, Phys. Rev. Lett. , 2933(1998); Phys. Rev. B , 1149 (2000). S. Yan, D. A. Huse, and S. R. White, Science , 1173(2011). B. Bernu, P. Lecheminant, C. Lhuillier, and L. Pierre,Phys. Rev. B , 10048 (1994). W. LiMing, G. Misguich, P. Sindzingre, and C. Lhuillier,Phys. Rev. B. , 6372 (2000). H. Kawamura and S. Miyashita, J. Phys. Soc. Jpn. , 9(1984). H. Kawamura, A. Yamamoto, and T. Okubo, J. Phys. Soc.Jpn. , 023701 (2010). S.-H. Lee, C. Broholm, T. H. Kim, W. Ratcliff, and S.-W.Cheong, Phys. Rev. Lett. , 3718 (2000). H. Ueda, H. Aruga-Katori, H. Mitamura, T. Goto, and H.Takagi, Phys. Rev. Lett. , 047202 (2005). K. Penc, N. Shannon, and H. Shiba, Phys. Rev. Lett. ,197203 (2004). F. Wang and A. Vishwanath, Phys. Rev. Lett. , 077201(2008). H. Yoshida, Y. Muraoka, T. S¨orgel, M. Jansen, and Z.Hiroi, Phys. Rev. B , 020408(R) (2006). H. Yoshida, S. Ahlert, M. Jansen, Y. Okamoto, J. Ya-maura, and Z. Hiroi, J. Phys. Soc. Jpn. , 074719 (2008). H. Nozaki, J. Sugiyama, M. Janoschek, B. Roessli, V. Pom-jakushin, L. Keller, H. Yoshida, and Z. Hiroi, J. Phys.:Condens. Matter , 104236 (2008). S. Ji, E. J. Kan, M.-H. Whangbo, J.-H. Kim, Y. Qiu, M.Matsuda, H. Yoshida, Z. Hiroi, M. A. Green, T. Ziman,and S.-H. Lee, Phys. Rev. B , 094421 (2010) H. Yoshida, E. Takayama-Muromachi, and M. Isobe, J.Phys. Soc. Jpn , 123703 (2011). J. Rodriguez-Carvajal, Physica B , 55 (1993). F. Ye, Y. Ren, Q. Huang, J. A. Fernandez-Baca, P. Dai,J. W. Lynn, and T. Kimura, Phys. Rev. B , 220404(R)(2006). N. Terada, S. Mitsuda, H. Ohsumi, and K. Tajima, J. Phys.Soc. Jpn. , 023602 (2006). M. Mekata, N. Yaguchi, T. Takagi, T. Sugino, S. Mitsuda,H. Yoshizawa, N. Hosoito, and T. Shinjo, J. Phys. Soc.Jpn. , 4474 (1993). T. Takagi and M. Mekata, J. Phys. Soc. Jpn. , 4609(1995). R. S. Fishman, F. Ye, J. A. Fernandez-Baca, and J. T.Haraldsen, and T. Kimura, Phys. Rev. B , 140407(R)(2008). W. B. Yelon, D. E. Cox, and M. Eibsch¨utz, Phys. Rev. B , 5007 (1975). M. Mekata and K. Adachi, J. Phys. Soc. Jpn. , 806(1978). Y. Nishiwaki, T. Nakamura, A. Oosawa, K. Kakurai, N.Todoroki, N. Igawa, Y. Ishii, and T. Kato, J. Phys. Soc.Jpn. , 104703 (2008). S. Niitaka, H. Kageyama, K. Yoshimura, K. Kosuge, S.Kawano, N. Aso, A. Mitsuda, H. Mitamura, and T. Goto,J. Phys. Soc. Jpn.70