1Department of Physics, Advanced Sciences, G.S.H.S. Ochanomizu University, Bunkyo, Tokyo 112-8610, Japan2RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan3Neutron Science Laboratory, Institute for Solid State Physics, The University of Tokyo, Tokai, Ibaraki 319-1106, Japan4Department of Applied Physics, Tokyo University of Science, Katsushika, Tokyo 125-8585, Japan5Neutron Technologies Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6475, U.S.A.6Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.A.
Received March 7, 2021; Revised May 7, 2021; Accepted May 11, 2021; Published June 8, 2021
The magnetic diffuse scattering of YBaCo4O7 and LuBaCo4O7 with alternating kagome and triangular lattices has been studied through neutron scattering measurements on single-crystal samples. In YBaCo4O7, the line-shape-magnetic-diffuse scattering is observed around the magnetic transition temperature, which is consistent with the Monte Carlo simulation. In LuBaCo4O7, the line-shape-magnetic-diffuse scattering has not been detected. At low temperature, the propagation vectors of the observed magnetic reflections are different between YBaCo4O7 and LuBaCo4O7. The structural distortion in RBaCo4O7 is suggested to play a critical role for the magnetic properties.
Spin systems on the pyrochlore, triangular, and kagome lattices are well-known examples of geometrically frustrated systems.1,2) The strong magnetic fluctuations are expected to induce various interesting magnetic properties. One of the geometrically frustrated systems is RBaCo4O7 (R = Ca, Y, and rare-earth elements).3,4) The RBaCo4O7 has a network of CoO4 tetrahedra, as shown in Fig. 1(a). The Co ions form alternate-stacking layers of the kagome and large triangular lattices. In Fig. 1(b), the layer of the kagome lattice and that of the large triangular lattice are depicted by blue and red colors, respectively. RBaCo4O7 with R3+ has three Co2+ and one Co3+ ions in a formula unit, but charge ordering has not been reported. The network of Co-ions in RBaCo4O7 is similar to that in the pyrochlore system, where the network of the corner-sharing tetrahedra forms alternating kagome and large triangular lattices along the [111] direction.2) In RBaCo4O7, the stacking structure of the kagome and the large triangular lattices induces a network of hexahedra in which two tetrahedra share faces, as shown in Fig. 1(b). The large antiferromagnetic interaction estimated from the magnetic susceptibility3,5) is expected to induce magnetic frustration.
Figure 1. (Color online) (a) Schematic crystal structure of RBaCo4O7. (b) Exchange interaction passes between the Co ions. \(J_{1}\) is the exchange interaction between the Co moments in the kagome lattice. \(J_{2}\) is the exchange interaction connecting kagome and triangular lattices. Co ions of the kagome and large triangular lattices are shown by the blue and red balls, respectively. (c) Reciprocal lattice space in \((H\ K\ 0)\)-plane. Closed circles, opened squares, and opened circles indicate the fundamental, \(\boldsymbol{Q}_{1/2}\)-, and \(\boldsymbol{Q}_{1/3}\)-points. The hexagon shape (LSMDS) is highlighted by blue solid lines. Red broken and green dotted lines represent the directions of scans in Figs. 2(f) and 3(f), respectively.
In our previous study of YBaCo4O7, two transitions were found at temperatures \(T_{\textit{Yc}1}=70\) K and \(T_{\textit{Yc}2}=105\) K.6,7) Below \(T_{\textit{Yc}2}\), the magnetic reflections were observed at the \(\boldsymbol{Q}_{1/2}\)-points, \((h_{0}\pm 1/2,k_{0},l_{0})\), \((h_{0},k_{0}\pm 1/2,l_{0})\), \((h_{0}\pm 1/2,k_{0}\pm 1/2,l_{0})\), and the \(\boldsymbol{Q}_{1/3}\)-points, \((h_{0}\pm 1/3,k_{0}\pm 1/3,l_{0})\) (\(h_{0}\), \(k_{0}\), and \(l_{0}\) = integers). The \((H\ K\ 0)\)-plane map in Fig. 1(c) shows \(\boldsymbol{Q}_{1/2}\) (square) and \(\boldsymbol{Q}_{1/3}\) (circle) superlattice reflections with the fundamental Bragg reflection (filled circle). By connecting \(\boldsymbol{Q}_{1/2}\) and \(\boldsymbol{Q}_{1/3}\) reflections, they form a hexagon shape and surround the fundamental Bragg reflection at their center. In the neutron diffraction experiment performed by Manuel et al., the line-shape-magnetic-diffuse scattering was discovered just above \(T_{\textit{Yc}2}\) along the blue solid lines in the c-plane in Fig. 1(c).8) Throughout this paper, we describe such characteristic magnetic diffuse scattering as line-shape-magnetic-diffuse scattering (LSMDS). Since no temperature dependence of LSMDS was studied in Ref. 8, the detailed behavior of LSMDS is still to be reported. The magnetic transition at \(T_{\textit{Yc}2}\) is also accompanied by a structural transition from orthorhombic \(Pbn2_{1}\) to monoclinic \(P2_{1}\).9) Consequently, the \(\boldsymbol{Q}_{1/2}\) superlattice reflections include both nuclear and magnetic components in the low temperature phase below \(T_{\textit{Yc}2}\). Although the nature of the magnetic transition at \(T_{\textit{Yc}1}\) is unclear, it has been reported that intensities of \(\boldsymbol{Q}_{1/2}\) and \(\boldsymbol{Q}_{1/3}\) reflections rapidly increase with decreasing temperature T. Note that both transitions at \(T_{\textit{Yc}1}\) and \(T_{\textit{Yc}2}\) exhibit anomalies in the specific heat.6)
The purpose of this study is to explore the behavior of magnetic diffuse scattering, LSMDS, near \(\boldsymbol{Q}_{1/2}\) and \(\boldsymbol{Q}_{1/3}\) reflections discovered in YBaCo4O7 with the kagome and triangular lattices. In many geometrically frustrated systems, characteristic patterns of magnetic diffuse scattering have been reported,10,11) and unique magnetic models, for example, the spin-ice model, have been proposed to understand such patterns. We would like to note that the shape of the LSMDS in the YBaCo4O7 system is unique, and is distinctly different from previously reported patterns in other geometrically frustrated systems. Concomitantly, it is worth exploring the exact origin of LSMDS by examining the behavior of magnetic diffuse scattering and the magnetic superlattice reflections in RBaCo4O7 in more detail.
In this paper, neutron diffraction and Monte Carlo simulation have been performed on YBaCo4O7 and LuBaCo4O7 single crystal samples to examine the magnetic diffuse scattering of RBaCo4O7 with triangular and kagome lattices. From this study, we conclude that, despite the similar crystal structure, YBaCo4O7 and LuBaCo4O7 systems show qualitatively different magnetic behavior near the magnetic transition temperature ∼105 K. In YBaCo4O7, LSMDS has been observed just above \(T_{\textit{Yc}2}\), which is consistent with the Monte Carlo simulation. At low T, the growth of the magnetic peaks at \(\boldsymbol{Q}_{1/2}\)- and \(\boldsymbol{Q}_{1/3}\)-points was observed in the neutron diffraction while the Monte Carlo simulation gives only the \(\boldsymbol{Q}_{1/3}\) magnetic peaks. In LuBaCo4O7, LSMDS has not been detected near or above the first-order magnetic transition of \(T_{\textit{Lc}1}=105\) K, but broad incommensurate peaks have been observed at lower T. The difference in the magnetic behavior in two compounds and its relation to the structural transition will be presented in this paper.
2. Experimental Details
The YBaCo4O7 and LuBaCo4O7 single crystals were grown by the floating zone method. Y2O3, Lu2O3, BaCO3, and Co3O4 were mixed in the proper molar ratios. The mixtures were pressed into rods and sintered at 1000 °C. By using the obtained rod, the YBaCo4O\(_{7+\delta}\) and LuBaCo4O\(_{7+\delta}\) single crystals were grown in air. As reported in our previous studies, the YBaCo4O\(_{7+\delta}\) and LuBaCo4O\(_{7+\delta}\) single crystals were annealed in nitrogen and air, respectively, to adjust the oxygen content to \(\delta=0\).12,13) The single crystals were checked not to have an appreciable amount of impurity phases by powder X-ray measurement. The sample alignment has been performed by using the high-energy X-ray Laue system installed in ISSP, The University of Tokyo.14)
The neutron measurement on YBaCo4O7 was performed by using the cold-neutron chopper spectrometer CNCS installed at SNS, U.S.A.15,16) The incident neutron energy \(E_{\text{i}}\) was 15.1 meV, and the energy resolution at the elastic position was about \(dE/E=6.5\)%. In this paper, we focused only on elastic neutron study. The sample was cooled using a liquid helium cryostat. The neutron measurement on LuBaCo4O7 was performed by using the elastic diffuse scattering spectrometer CORELLI installed at SNS, U.S.A.17) A white neutron beam with incident neutron energy from 10 to 200 meV was used. The sample was cooled using the closed-cycle refrigerator. For both experiments, single crystals were oriented with the [100] and [010] axes in the horizontal plane. Throughout this paper, we use the hexagonal unit cell with the space group \(P6_{3}mc\). The lattice parameters in YBaCo4O7 are \(a = b = 6.308\) Å and \(c = 10.260\) Å, and those in LuBaCo4O7 are \(a = b = 6.283\) Å and \(c = 10.315\) Å. The \((-1,2,0)\) vector is used as the axis perpendicular to both \(a^{*}\)- and \(c^{*}\)-axes, as shown in Fig. 1(c). By rotating the sample around the vertical c-axis direction, we are measuring the diffraction pattern in the \((H, K, 0)\) scattering plane. The neutron intensities with finite \(c^{*}\)-components were measured simultaneously in the vertical direction of position-sensitive detectors.
3. Results
YBaCo4O7
Figure 2(a) shows the contour plot of neutron intensity in YBaCo4O7 measured at 10 K in the \((H\ K\ 0)\)-plane. Superlattice reflections were observed at both \(\boldsymbol{Q}_{1/2}\)- and \(\boldsymbol{Q}_{1/3}\)-points depicted in Fig. 1(c) in addition to fundamental Bragg reflections and Al-powder lines. The profiles of the observed superlattice reflection were broad, i.e., not resolution-limited at the lowest temperature \(T=10\) K measured in this study, indicating that YBaCo4O7 exhibits no long-range-order down to 10 K. The broad peaks at \(\boldsymbol{Q}_{1/2}\)- and \(\boldsymbol{Q}_{1/3}\)-points were observed in our previous results,6) while Khalyavin et al. did not detect the magnetic signal at the \(\boldsymbol{Q}_{1/3}\)-points.9)
Figure 2. (Color online) Elastic neutron scattering intensities of YBaCo4O7 measured in CNCS spectrometer. (a)–(d) Contour plots of elastic neutron intensities in the \((H\ K\ 0)\)-plane at (a) 10 K, (b) 80 K, (c) 110 K, and (d) 300 K. (e) Contour plot of the elastic neutron intensity measured at 110 K for the \((\text{1.5-$K'$},2K',L)\). (f) Elastic neutron intensities at the \((\text{1.5-$K'$},2K',0)\) for four T values.
The contour plots of the selected temperatures measured at 80, 110, and 300 K are shown in Figs. 2(b), 2(c), and 2(d), respectively. In the contour plot at 110 K just above \(T_{\textit{Yc}2}\) shown in Fig. 2(c), the characteristic line-shape-magnetic-diffuse scattering is clearly observed along the blue-lines depicted in Fig. 1(c), and we name such magnetic diffuse scattering as LSMDS. This observation is consistent with what Manuel et al. reported in Ref. 8.
To examine the magnetic correlation along the c-axis direction, the contour plot measured at 110 K on the \((\text{1.5-$K'$},2K',L)\)-plane is shown in Fig. 2(e). The profile of LSMDS is broad within the ab-plane, while it is sharp along the c-axis. This means that the magnetic correlation is long-ranged along the c-axis.
The one-dimensional (1D) profiles of the elastic neutron intensities at 10, 80, 110, and 300 K along \((\text{1.5-$K'$},2K',0)\), which correspond to the red broken line in Fig. 1(c), are shown in Fig. 2(f). Magnetic diffuse scattering has not been detected at 300 K. At 110 K, LSMDS has the largest intensity. With further decreasing T, the intensity of LSMDS decreases. The superlattice reflections and LSMDS coexist at 80 K. At 10 K, broad magnetic diffuse scattering has been observed at \(\boldsymbol{Q}_{1/2}\)- and \(\boldsymbol{Q}_{1/3}\)-points. From the T dependence of the neutron intensities around \(K'=0.25{\text{–}}0.30\) in Fig. 2(f), we found that the neutron intensity at 80 K is smaller than at 10 and 110 K. This indicates that two kinds of magnetic diffuse scattering, LSMDS and broad superlattice reflections, co-exist. LSMDS has the largest intensity around \(T_{\textit{Yc}2}\), and decreases with decreasing T. By contrast, the broad superlattice reflections at \(\boldsymbol{Q}_{1/2}\)- and \(\boldsymbol{Q}_{1/3}\)-points are weak around \(T_{\textit{Yc}2}\), and their intensities increase with decreasing T. The broad peaks of the superlattice reflections seem to form a hexagon shape at low T.
LuBaCo4O7
In this subsection, we present the behavior of the magnetic diffuse scattering observed in LuBaCo4O7, the structurally isomorphic reference compound to YBaCo4O7. Figure 3(a) shows the contour plot of neutron intensity at 6 K in the \((H\ K\ 0)\)-plane of LuBaCo4O7. The superlattice reflections are observed around \(\boldsymbol{Q}_{1/2}\)-points and at \(\boldsymbol{Q}_{1/3-\delta}=\boldsymbol{Q}_{1/3}\pm\delta\boldsymbol{q}\), where \(\delta\boldsymbol{q}=(\delta,\delta,0)\), \((2\delta,-\delta,0)\), and \((-\delta,2\delta,0)\). The δ value is about 0.07 at 6 K.18) The arrangement of \(\boldsymbol{Q}_{1/2}\) and \(\boldsymbol{Q}_{1/3-\delta}\) superlattice peaks forms a clear hexagon shape similar to those in YBaCo4O7.
Figure 3. (Color online) Elastic neutron scattering intensities of LuBaCo4O7 measured in CORELLI spectrometer. (a)–(e) Contour plots of elastic neutron intensities in the \((H\ K\ 0)\)-plane at (a, b) 6 K, (c) 45 K (d) 80 K, and (e) 98 K with increasing temperature. (b) Contour plot around \((1.5,0,0)\) with an enlarged scale. (f) Elastic neutron intensities at the \((H,H,0)\) for five T values with increasing temperature.
Figure 3(b) shows the expanded map around \((1.5,0,0)\) at 6 K. The reflection around \((1.5,0,0)\) has a characteristic shape. In addition to \(\boldsymbol{Q}_{1/2}\)- and \(\boldsymbol{Q}_{1/3-\delta}\)-points, four satellite reflections can be seen around \((1.5,0,0)\), which are indicated by arrows in Fig. 3(b). These superlattice reflections may be incommensurate satellites of \(\boldsymbol{Q}_{1/2}\), being denoted as \(\boldsymbol{Q}_{1/2}\pm\delta\boldsymbol{q}\). However, we cannot determine the precise magnetic propagation vector of these satellite reflections around \((1.5,0,0)\).
The contour plots of the neutron intensities measured at 45, 80, and 98 K with increasing T are shown in Figs. 3(c), 3(d), and 3(e), respectively. The T dependence of the 1D profiles of the elastic neutron intensities along \((H,H,0)\), which corresponds to the green dotted line in Fig. 1(c), is shown in Fig. 3(f). With increasing T, the δ value of the incommensurate \(\boldsymbol{Q}_{1/3-\delta}\)-points decreases, and becomes almost zero around \(T_{\textit{Lc}1}=105\) K. LuBaCo4O7 exhibits a first-order magnetic transition at \(T_{\textit{Lc}1}\), and the magnetic reflections disappear above \(T_{\textit{Lc}1}\), at which the structural distortion occurs with accompanying a large change in the lattice constants.13) It should be noted that, unlike the case of YBaCo4O7, LSMDS could not be observed either near or above \(T_{\textit{Lc}1}\). Corresponding to this transition, the superlattice reflections around \(\boldsymbol{Q}_{1/2}\) have both nuclear and magnetic components below \(T_{\textit{Lc}1}\), but have only the nuclear component above \(T_{\textit{Lc}1}\). This nuclear component at the \(\boldsymbol{Q}_{1/2}\)-points survives up to the structural transition at \(T_{\textit{Lc}2}=165\) K.
4. Analysis and Discussion
To understand the magnetic diffuse scattering in RBaCo4O7, Monte Carlo simulation has been performed. Our previous inelastic neutron result on YBaCo4O7 could be explained by a Heisenberg spin Hamiltonian, where the magnitude of the exchange interaction within the kagome lattice, \(J_{1}\), is equal to that connecting kagome and triangular lattices, \(J_{2}\)12) [Fig. 1(b)]. In the Monte Carlo simulation, the Heisenberg spin Hamiltonian without the single-ion anisotropy is applied. The magnitudes of the antiferromagnetic interactions within the kagome lattice and connecting kagome and triangular lattices are set to be J (\(=J_{1}=J_{2}\)). In the calculation a \(16\times 16\times 8\) supercell was used. Since 8 magnetic sites exist in the hexagonal unit cell, there are 16384 sites in the supercell. RBaCo4O7 has three Co2+ and one Co3+ ions, but no clear charge ordering has been reported. We assumed that the magnitudes of all magnetic moments are equal. In the simulation, a combination of the heat bath method, the over-relaxation method, and the temperature replica exchange method has been applied. The calculation conditions of this Monte Carlo simulation are similar to those of the previous simulation reported by Manuel et al.8)
Figures 4(a), 4(b), and 4(c) show the contour plots of the magnetic structure factors, \(S_{\mathbf{q}}\), in the \((H\ K\ 0)\)-plane calculated by Monte Carlo simulation with the \(J_{1}=J_{2}\) (\(=J\)) model at \(T/J = 0.20\), 0.42, and 0.62, respectively. At \(T/J = 0.62\), in Fig. 4(c), the \(\boldsymbol{Q}\)-regions around ferromagnetic points, such as \((0,0,0)\) and \((2,0,0)\), have little intensity and exhibit “the intensity hole”. With decreasing T to \(T/J = 0.42\) shown in Fig. 4(b), the intensity holes expand their region, and another intensity hole appears at the \((1,1,0)\) position. Surrounding the intensity holes, a network of magnetic diffuse scattering is formed, and this network seems to trace the blue-lines shown in Fig. 1(c). With further decreasing T, the magnetic intensities are centered at \(\boldsymbol{Q}_{1/3}\)-points, as shown in Fig. 4(a).
Figure 4. (Color online) Static structure factor obtained from Monte Carlo simulation with \(J_{1}=J_{2}\) model. (a)–(c) Contour plots of the calculated magnetic structure factors in the \((H\ K\ 0)\)-plane at \(T/J\) = (a) 0.20, (b) 0.42, and (c) 0.62. (d) Contour plot of the calculated magnetic structure factor at \(T/J=0.42\) in the \((\text{1.5-$K'$},2K',L)\).
Figure 4(d) shows the contour plot of the calculated \(S_{\mathbf{q}}\) in the \((\text{1.5-$K'$},2K',L)\) at \(T/J=0.42\). The magnetic diffuse scattering is broad within the c-plane, but is well-concentrated around \(L\sim 0\), indicating that the magnetic correlation along the c-axis is long-ranged.
Now we compare the Monte Carlo results with the experimental observation. The Monte Carlo simulations at \(T/J = 0.42\) shown in Figs. 4(b) and 4(d) well reproduce the features of experimentally observed LSMDS in YBaCo4O7 shown in Figs. 2(c) and 2(e). Both simulation and experimental observation demonstrate that the spin correlation is extremely anisotropic, and is short-ranged in the ab-plane, but nearly long-ranged along the c-axis. In other words, the spins form a three-dimensional network on the kagome and triangular lattice, which accompanies with unique geometrical frustrations.
At lower T at \(T/J = 0.20\), our simulation predicts that the superlattice reflections appear at the \(\boldsymbol{Q}_{1/3}\)-points. In the experiments, however, YBaCo4O7 shows superlattice reflections at \(\boldsymbol{Q}_{1/3}\)- and \(\boldsymbol{Q}_{1/2}\)-positions, as shown in Fig. 2(a). Furthermore, the \(\boldsymbol{Q}_{1/3}\)- and \(\boldsymbol{Q}_{1/2}\)-peaks become incommensurate in LuBaCo4O7 as shown in Figs. 3(a) and 3(b). Nevertheless, the hexagon networks similar to the simulation in Fig. 4(a) are observed in both YBaCo4O7 and LuBaCo4O7 compounds. Note that the observation of the \(\boldsymbol{Q}_{1/2}\) superlattice reflections should be attributed to the influence of the structural transitions accompanied by the magnetic transitions in both compounds.
As mentioned above, at the magnetic transition temperatures \(T_{\textit{Yc}2}\) (\(=105\) K) and \(T_{\textit{Lc}1}\) (\(=105\) K) in YBaCo4O7 and LuBaCo4O7, structural transitions also occur. For YBaCo4O7, the structure changes from orthorhombic to monoclinic at \(T_{\textit{Yc}2}\). The monoclinic distortion increases gradually with decreasing T,9) indicative of the spin–lattice coupling. In LuBaCo4O7, a first-order structural transition has been reported at \(T_{\textit{Lc}1}\), where the lattice constant shows a discontinuous change. A similar discontinuity in magnetic susceptibility has been also observed.13,19) Such discontinuity naturally explains the lack of LSMDS in the LuBaCo4O7 compound above \(T_{\textit{Lc}1}\).
The T dependence of the magnetic diffuse scattering in YBaCo4O7 indicates a possibility that LSMDS is related to the Z2 vortex transition, which is characterized by chirality.20) For the triangular lattice, the Z2 topological transition has been theoretically proposed. LSMDS connecting the magnetic Bragg points is expected to appear around the Z2-vortex transition temperature, and its intensity reaches the maximum at the Z2-vortex transition. Furthermore, the spin correlation length remains finite even at low T because the degree of freedom of the chirality blocks the spin ordering. The behavior of the magnetic scattering proposed by the theoretical study is similar to the magnetic diffuse scattering observed in YBaCo4O7. LSMDS connecting the \(\boldsymbol{Q}_{1/3}\)-points with the \(\boldsymbol{Q}_{1/2}\)-points was observed in YBaCo4O7, and LSMDS has a large intensity at 110 K which is just above the magnetic transition temperature. Furthermore, no long-range-magnetic order exists even at low T, while the broad superlattice reflection is only observed. We note that the Z2-vortex transition has been proposed to occur on the triangular lattice, whereas RBaCo4O7 compounds consist of the kagome and large triangular lattices. Consequently, further studies are necessary to clarify the Z2-vortex order in RBaCo4O7 which has a three-dimensional network of magnetic correlation.
5. Conclusions
The magnetic diffuse scattering of the geometrical frustration systems YBaCo4O7 and LuBaCo4O7 was studied by using neutron scattering measurements and Monte Carlo simulation. In YBaCo4O7, the line-shape-magnetic-diffuse scattering appears around the magnetic transition temperature, and broad magnetic reflections were observed at \(\boldsymbol{Q}_{1/2}\)- and \(\boldsymbol{Q}_{1/3}\)-points at low temperature. The Monte Carlo simulation is almost consistent with the magnetic diffuse scattering in YBaCo4O7. In LuBaCo4O7, the line-shape-magnetic-diffuse scattering was not observed, and magnetic scattering with an incommensurate propagation vector was detected. Our results indicate that the structural distortion in RBaCo4O7 is closely related to the magnetic properties.
Acknowledgments
Travel expense for the neutron experiment on CNCS and CORELLI was supported by a General User Program for Neutron Scattering Experiments, Institute for Solid State Physics, The University of Tokyo (Proposal Nos. 13531, 13569, and 16909). The research at ORNL's Spallation Neutron Source was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy. This work was supported by KAKENHI (24740224 and 15K05123).
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