Universal magnetic structure of the half-magnetization phase in Cr-based spinels
M. Matsuda, K. Ohoyama, S. Yoshii, H. Nojiri, P. Frings, F. Duc, B. Vignolle, G. L. J. A. Rikken, L.-P. Regnault, S.-H. Lee, H. Ueda, Y. Ueda
aa r X i v : . [ c ond - m a t . m t r l - s c i ] J u l Universal magnetic structure of the half-magnetization phase in Cr-based spinels
M. Matsuda, K. Ohoyama, S. Yoshii, H. Nojiri, P. Frings, F. Duc, B.Vignolle, G. L. J. A. Rikken, L.-P. Regnault, S.-H. Lee, H. Ueda, and Y. Ueda Quantum Beam Science Directorate, Japan Atomic Energy Agency (JAEA), Tokai, Ibaraki 319-1195, Japan Institute for Materials Research, Tohoku University, Katahira, Sendai 980-8577, Japan Laboratoire National des Champs Magn´etiques Intenses,UPR3228 CNRS-INSA-UJF-UPS, Grenoble & Toulouse, France CEA-Grenoble, INAC-SPSMS-MDN, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France Department of Physics, University of Virginia, Charlottesville, VA 22904-4714, USA The Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan (Dated: May 29, 2018)Using an elastic neutron scattering technique under a pulsed magnetic field up to 30 T, wedetermined the magnetic structure in the half-magnetization plateau phase in the spinel CdCr O .The magnetic structure has a cubic P
32 symmetry, which is the same as that observed in HgCr O .This suggests that there is a universal field induced spin-lattice coupling mechanism at work in theCr-based spinels. PACS numbers: 75.30.Kz, 75.25.+z, 75.50.Ee
When an external magnetic field is applied on stronglyinteracting spin systems, novel collective phenomena mayarise.[1] A well-known example is the field-induced con-densation of magnons in quantum magnets.[2] Anotheris the field-induced fractional magnetization plateau ob-served in frustrated magnets.[3] Interest on the lattersystem stems from the degenerate ground state due tothe triangular motif of the magnetic lattice.[4, 5] For in-stance, for a tetrahedron made of four isotropic classicalspins, any spin configuration with total zero spin can bethe ground state. There is an infinite number of such con-figurations that satisfy the criterion. When such tetrahe-dra are arranged in a corner-sharing network, sometimescalled the pyrochlore lattice, the ground state degeneracybecomes macroscopic, and exotic magnetic properties areexpected at low temperatures.[6, 7, 8] If the spin degreeof freedom is coupled with the lattice or orbital degreeof freedom, the system can undergo a phase transition atlow temperature into a crystallographically distorted andmagnetic ordered state.[9, 10, 11, 12, 13, 14, 15, 16, 17]When an external magnetic field ( H ) is applied to the or-dered state, the fractional magnetization phase appears,and is associated with each tetrahedron having major-ity (minority) spins aligned parallel (antiparallel) to H .There can be many ways of organizing the majority andminority spins over the entire lattice, and how a certainstructure can be stabilized over a wide range of H andwhether or not an universal ground state for each typeof frustrated lattice and Hamiltonians exists are of issue.The Cr-based spinel A Cr O ( A = Hg, Cd, and Zn) isan ideal model system with the simplest and most frus-trating spin Hamiltonian. It has a cubic F d ¯3 m crys-tal structure at room temperature where the magneticCr ( t g ; s = 3 /
2) ions without orbital degeneracy formthe pyrochlore lattice, and due to the edge-sharing net-work of CrO octahedra, the nearest neighbor exchange interactions are dominant. Experimentally, it has beenshown that despite the strong magnetic interactions evi-denced by their large Curie-Weiss temperatures, Θ CW = −
32 K (Hg), −
88 K (Cd), and −
390 K (Zn), the systemremains in a spin-liquid state down to temperatures ( T )much lower than | Θ CW | , indicating the presence of strongfrustration.[3] Upon further cooling, it undergoes a tran-sition at 6 K (Hg), 7.8 K (Cd), and 12.5 (Zn) due to spin-lattice coupling into a magnetically long range orderedstate. [9, 10, 18] The nature of the magnetic structure andlattice distortion is different for different A site ions: thesymmetry of the low T crystal structure is orthorhombic F ddd , tetragonal I /amd , and tetragonal I ¯4 m Q m = (1,0,1/2), (1,0,0) for Hg [18],a single incommensurate Q m = (0 , δ,
1) for Cd [10], andtwo commensurate Q m = (1/2,1/2,0), (1,0,1/2) for Zn[19]. These indicate that the Cr spinels are very close toa critical point surrounded by several different spin struc-tures in phase space, and the microscopic mechanism ofthe zero-field spin-lattice coupling depends on the deli-cate balancing acts between spin and lattice degrees offreedom that vary with the A site ion.When an external field, H , is applied, the Cr spinelundergoes a phase transition into a half-magnetizationplateau phase at H c = 10 T for Hg, 28 T for Cd, and120 T for Zn [3, 20, 21], suggesting that each tetrahe-dron has three up (majority) and one down (minority)spins (3:1 constraint). The field-induced magnetic andchemical structure of HgCr O were determined to havethe P
32 symmetry or its mirror image P
32 since its H c is within the steady magnet capability available atneutron facilities.[18] A question that arises is whether ornot the nature of the field-induced phase in the Cr spinelsvaries with different A ions as it does for the zero-field FIG. 1: (Color online) (a) A schematic diagram of the [111]and [1¯10] horizontal scattering plane that was used for ourneutron scattering measurements. The external pulsed mag-netic field, H , was applied horizontally 7 ◦ away from the [111]direction as shown by an arrow. The open circles representcommensurate wave vector positions, and the filled circlesrepresent the (0, δ ,1) incommensurate magnetic Bragg posi-tions observed for zero magnetic field. (b) Our elastic neu-tron scattering data taken with H = 0 at 2 K along the(1.04+ h , − h , h ) as shown by an arrow in (a). spin-lattice coupling. Studying other Cr spinels has how-ever been impossible because of their high critical fieldsthat are beyond the current steady magnet technologyavailable at neutron facilities. Very recently, a pulsedmagnet capability that can go up to 30 T has been imple-mented at neutron facilities, opening up a new researchopportunity in this field.Here, we report our elastic neutron scattering mea-surements on CdCr O under the pulsed magnetic fieldto study its half-magnetization plateau phase. We showthat for H > H c = 28 T the incommensurate mag-netic peaks disappear while new peaks appear with thecharacteristic wave vector of Q m =(1,0,0) but not at the(2,¯2,0) point. This clearly demonstrates that the half-magnetization plateau phase of CdCr O has the same P
32 magnetic structure as that of HgCr O . Our re-sults suggest that the observed P
32 state might be thegeneric ground state of the field-induced phase of theCr-spinels, despite their different crystal and magneticstructures observed at H = 0.A single crystal that has a shape of a thin plate ( ∼ × × ∼
40 mg was used. Sincenatural Cd has a large neutron absorption cross section,a single crystal enriched with
Cd isotope was usedfor our measurements. The elastic neutron scatteringexperiments were carried out on the thermal neutrontriple-axis spectrometers IN22 at Institut Laue-Langevin(ILL). The incident and final neutron energies were fixedto E i =34.8 meV. Contamination from higher-order neu-trons was effectively eliminated by a PG filter. The 40mg single crystal was mounted with the [111] and [1¯10]axes in the horizontal scattering plane. A small magnetcoil made of CuAg wire was mounted surrounding the C o un t s / pu l s e / m s m ag n e t i c f i e l d ( T ) (1.0675, -1.0125, 0.0275) (a) m ag n e t i c f i e l d ( T ) C o un t s / pu l s e / m s (1, -1, 0) (b) FIG. 2: (Color online) Time dependence of the magnetic field(solid red lines) and neutron counts (filled circles) measured at(1.0675, − −
1, 0) reflections at the ini-tial temperature T =2.5 K. The corresponding magnetic fieldis shown on the right y-axis. The measurements were per-formed over 200 and 150 magnetic field pulses for (1.0675, − −
1, 0) reflections, respectively, andthe data were summed. Binning times were 200 and 80 µ sfor (1.0675, − −
1, 0) reflections, re-spectively. The vertical dotted lines are drawn at the timescorresponding nominally to the critical field, H c = 28 T. Thehorizontal dashed lines represent the background levels de-termined at wave vectors away from the reflection positions.The gray thick lines are guide for eyes. crystal on a cryostat insert and the apparatus was cooledin a standard He cryostat. [22, 23, 24] The magnet coilin the cryostat was connected to a transportable capac-itor bank that resided outside the cryostat. A half-sineshaped pulsed magnetic field of 8 msec duration was gen-erated by using a capacitor bank.[25] The magnetic fieldwas measured by a set of pick-up coils installed aroundthe sample. The magnet coil limited the accessible scat-tering angle below 30 ◦ , which allowed us to reach onlytwo commensurate reflections at (1, − − − O shows a spiral mag-netic order with a single characteristic wave vector of Q m = (0, δ , 1) or ( δ ,0,1) where δ ∼ .
09, accompaniedby a tetragonal distortion below T N = 7 . ◦ are (1, − ± δ ,0) and(1 ± δ, − ∼ − , they could bedetected in our measurements because the full-width-of-the-half-maximum of the vertical instrumental resolutionwas 0.144 ˚A − . We performed elastic scans around the(1, − c o un t s / pu l s e / m s magnetic field (T) (a) (1.0675, -1.0125, 0.0275) field_up field_down c o un t s / pu l s e / m s magnetic field (T) (b) (1, -1, 0)6040200 c o un t s / pu l s e / m s magnetic field (T) (c) (2, -2, 0) FIG. 3: (Color online) Magnetic field dependence of the peakintensity of the (a) (1.0675, − −
1, 0),and (c) (2, −
2, 0) reflections measured at T =2.5 K with theascending (filled circles) and descending (open circles) field.For better statistics, binning times were 200, 80, and 80 µ s for(1.0675, − −
1, 0), and (2, −
2, 0) reflec-tions, respectively. The vertical dotted lines are drawn at thetimes corresponding nominally to the critical field, H c = 28 T.The horizontal dashed lines represent background intensity. sitions shown in Fig. 1 (a). In Fig. 1 (b), a typical elasticscan taken along the (1,1,1) direction centered at (1.04, − k and k of Fig. 1 (a). Thefour peak positions, k , k , k , and k , correspond tothe IC peaks at (1 . , -1 , , (1 , -1 . , , (1 , -0 . , , and(0 . , -1 ,
0) projected on the scattering plane.Figure 2(a) shows the time dependence of the elasticneutron scattering intensity measured at the IC magneticpeak of (1.0675, − H = 8 T, and remains constant at 20counts per 200 pulses up to 3 ms, after which the in-tensity sharply decreases to background level. The 3 mstime corresponds to H = 28 T, which is consistent withthe critical field observed in the previous bulk magneti-zation measurements[3]. The magnetic field reaches themaximum 29.6 T at 3.9 ms after which H decreases. TheIC magnetic signal remains zero up to 4.6 ms ( H = 28 T)after which the intensity increases back to the interme-diate level at H = 8 T but not to the original intensity FIG. 4: Magnetic structures with cubic P
32 (a) and rhom-bohedral R ¯3 m (b) symmetries. Open and filled circles repre-sent up and down spins, respectively. Each tetrahedron hasthree up and one down spins. at H = 0 T. The hysteretic behavior originates from themagnetic domain orientation and is consistent with theprevious result. [26] While waiting about 8 minutes forthe next current pulse, the sample was warmed up to 20K > T N and cooled back down to 2.5 K to recover theoriginal zero-field intensity.In order to find out where the elastic magnetic inten-sity that disappeared at the IC wave vector for fields H >
28 T transferred to, we performed similar measure-ments at a commensurate Q =(1 , − ,
0) position. Fig-ure 2(b) and Figure 3(b) show the results as a functionof time and field. When a magnetic field was injected,no signal was initially observed at (1, − O enters the half-magnetization plateau state upon application of an exter-nal magnetic field, the magnetic structure changes fromthe incommensurate spiral to a commensurate collinearspin structure with Q m =(1,0,0). Once the 3:1 constraintis imposed, the (100)-type reflections are consistent withtwo non-equivalent spin configurations for the pyrochlorelattice: one with cubic P
32 symmetry (Fig. 4 (a)) andthe other with rhombohedral R ¯3 m symmetry (Fig. 4(b)).[18] To distinguish between the two models, we alsoperformed similar pulsed field measurements at (2 , ¯2 , R ¯3 m structure should produce mag-netic Bragg scattering while the P
32 structure wouldnot. At H =0, nuclear Bragg intensity is observed at(2 , ¯2 , , ¯2 ,
0) intensity doesnot change as the system enters the half-magnetizationphase. Thus, we conclude that the half-magnetizationspin state of CdCr O has the same P
32 spin structureas observed in HgCr O . This is rather surprising be-cause the two systems have quite different ground statesat H = 0.Why is the P
32 spin structure favored as the groundstate of the half-magnetization phase in Cr-spinels, re-gardless of their different zero field ground states? Thereare many ways of arranging the pyrochlore lattice withtetrahedra holding the 3:1 constraint because there isconsiderable freedom in choosing the location of the downspin on each tetrahedron. The most relevant Hamiltonianfor the Cr-spinels is the nearest neighbor exchange inter-action that is sensitive to the bond distance minus anelastic energy associated with the displacements of themagnetic atoms. This effective hamiltonian has alreadybeen investigated theoretically as an Einstein phononmodel, showed that maximizing the displacements orminimizing the Hamiltonian occurs in a unique bendingpattern of tetrahedra that has the P
32 symmetry.[27]Previous combined neutron and synchrotron x-ray mea-surements on HgCr O showed that at the field-inducedphase transition the crystal structure indeed becomes cu-bic with the P
32 symmetry.[18] A recent synchrotronx-ray diffraction experiments on CdCr O under pulsedmagnetic field showed that the crystal structure of thehalf-magnetization phase is cubic as well. [28] These re-sults suggest that the simple effective Hamiltonian de-scribes the physics of the field-induced phase transitioninto the half-magnetization plateau phase in the Cr-spinels.In summary, our elastic neutron scattering experi-ments on a single crystal of CdCr O under a pulsedmagnetic field up to 30 T showed that the Cr-spinels, A Cr O with nonmagnetic A ions have a unique field-induced half-magnetization state, regardless of their dif-ferent zero-field ground states. This is consistent withthe theoretical prediction based on the simplest Hamilto-nian for the spin-lattice coupling with the nearest neigh-bor exchange interaction and an elastic energy term.[27]In addition, our study demonstrates that with the newpulsed magnet capability up to 30 T now available atneutron facilities, new research opportunities in the fieldof frustrated magnetism open up.This work was partially supported by Grant-in-Aid forScientific Research on priority Areas ”High Field SpinScience in 100T” (No.451), ”Novel States of Matter In-duced by Frustration” (19052004 and 19052008) fromthe Ministry of Education, Culture, Sports, Science andTechnology (MEXT) of Japan, and by ICC-IMR centerof Tohoku University. S.H.L. is supported by U.S. DOE through DE-FG02-07ER46384. [1] S. Sachdev, Nature Phys. , 173 (2008).[2] T. Nikuni, M. Oshikawa, A. Oosawa, and H. Tanaka,Phys. Rev. Lett. , 5868 (2000).[3] H. Ueda, H. Aruga-Katori, H. Mitamura, T. Goto, andH. Takagi, Phys. Rev. Lett. , 047202 (2005).[4] S. T. Bramwell and M. J. P. Gingras, Science , 24(2006).[6] R. Moessner and J. T. Chalker, Phys. Rev. Lett. , 2929(1998).[7] B. Canals and C. Lacroix, Phys. Rev. Lett. , 2933(1998); Phys. Rev. B , 1149 (2000).[8] S.-H. Lee et al. , Nature , 856 (2002).[9] S.-H. Lee, C. Broholm, T. H. Kim, W. Ratcliff, and S.-W.Cheong, Phys. Rev. Lett. , 3718 (2000).[10] J.-H. Chung et al. , Phys. Rev. Lett. , 247204 (2005).[11] Y. Yamashita and K. Ueda, Phys. Rev. Lett. , 4960(2000).[12] O. Tchernyshyov, R. Moessner, and S. L. Sondhi, Phys.Rev. Lett. , 067203 (2002).[13] D. I. Khomskii and T. Mizokawa, Phys. Rev. Lett. ,156402 (2005).[14] Y. Ueda, N. Fujiwara, and H. Yasuoka, J. Phys. Soc. Jpn. , 778 (1997).[15] S.-H. Lee et al. , Phys. Rev. Lett. , 156407 (2004).[16] H. Tsunetsugu and Y. Motome, Phys. Rev. B ,060405(R) (2003).[17] O. Tchernyshyov, Phys. Rev. Lett. , 157206 (2004).[18] M. Matsuda et al. , Nature Physics , 397 (2007).[19] S. Ji et al. , arXiv:0905.2127, to appear in Phys. Rev.Lett. (2009).[20] H. Ueda, H. Mitamura, T. Goto, and Y. Ueda, Phys.Rev. B , 094415 (2006).[21] E. Kojima, H. Ueda, Y. Ueda, S. Miyabe, and S.Takeyama, J. Phys: Conf. Ser. , 012023 (2009).[22] K. Ohoyama et al. , J. Phys.: Conf. Ser. 51, 506 (2006).[23] K. Ohoyama et al. , J. Magn. Magn. Mater. 310, e974(2007).[24] S. Yoshii et al. , to be published (2009).[25] P. Frings et al. , Rev, Sci. Instrum. 77, 063903 (2006).[26] M. Matsuda et al. , Phys. Rev. B , 104415 (2007).[27] D. L. Bergman, R. Shindou, G. A. Fiete, and L. Balents,Phys. Rev. B , 134409 (2006).[28] T. Inami et al. , J. Phys: Conf. Ser.51