Magnetism of the antiferromagnetic spin- dimer compound CrVMoO having an antiferromagnetically ordered state
We measured magnetization, specific heat, electron spin resonance, neutron diffraction, and inelastic neutron scattering of CrVMoO powder. An antiferromagnetically ordered state appears below K. We consider that the probable spin model for CrVMoO is an interacting antiferromagnetic spin- dimer model. We evaluated the intradimer interaction to be K and the effective interdimer interaction to be K. CrVMoO is a rare spin dimer compound that shows an antiferromagnetically ordered state at atmospheric pressure and zero magnetic field. The magnitude of ordered moments is . It is much smaller than a classical value . Longitudinal-mode magnetic excitations may be observable in single crystalline CrVMoO.
pacs:75.25.-j, 75.30.Cr, 75.40.Cx, 75.30.Ds
Two types of magnetic excitations exist in a magnetically ordered state. They are gapless transverse-mode (Nambu-Goldstone mode) Goldstone62 () and gapped longitudinal-mode (Higgs mode) Sachdev11 (); Podolsky11 () excitations corresponding to fluctuations in directions perpendicular and parallel to ordered moments, respectively. The transverse-mode (T-mode) excitations are well known as spin wave excitations. There are a few experimental observations on the longitudinal-mode (L-mode) excitations mainly because of their weak intensity. The L-mode excitations were observed in a pressure-induced or magnetic-field-induced magnetically ordered state of interacting antiferromagnetic (AF) spin- dimer compounds TlCuCl Kuroe08 (); Ruegg08 (); Merchant14 (); Matsumoto04 (); Matsumoto08a () and KCuCl Kuroe12 (). The ground state (GS) is a spin-singlet state at atmospheric pressure and zero magnetic field in these compounds. The ordered state is in the vicinity of quantum phase transition. Therefore, the L-mode excitations are observable because of large quantum fluctuations.
According to results of theoretical investigations, the L-mode excitations may be observed in an antiferromagnetically ordered state appearing on cooling at atmospheric pressure and zero magnetic field in interacting AF spin-cluster compounds Matsumoto10 (). A shrinkage of ordered magnetic moments by quantum fluctuations leads to a large intensity of the L-mode excitations. If the GS of the corresponding isolated spin cluster is a spin-singlet state, the shrinkage of ordered moments can be expected in an ordered state generated by the introduction of intercluster interactions.
In interacting spin clusters, the ordered state may appear under the condition that the value of an effective intercluster interaction is not so small compared with that of Matsumoto10 (). Here the effective intercluster interaction is given by the sum of the products of the absolute value of each intercluster interaction () and the corresponding number of interactions per spin () as . is the energy difference (spin gap) between the singlet GS and first-excited triplet states. It is advantageous for the appearance of the ordered state that is much smaller than the dominant intracluster interactions. In a spin- tetramer of which the Hamiltonian with and , the GS is a spin-singlet state and can be sufficiently small Hase97 (); Hase09 (); Hase16 (). The values of , , , and are 317, -162, 42, and 19 K, respectively, in CuCdBO Hase15 () and 240, -142, 30, and 17 K, respectively, in CuInVO Hase16 (). The ordered state appears in CuCdBO Hase09 (); Hase05 (); Hase15 () and CuInVO Hase16 () below the transition temperature and 2.7 K, respectively. Magnetic excitations in CuCdBO were studied by inelastic neutron scattering experiments on its powder Hase15 (). The results suggest the existence of the L-mode excitations. Magnetic excitations in CuInVO have not been investigated.
Spin dimer compounds are also attractive for investigation of the L-mode excitations at atmospheric pressure and zero magnetic field. In contrast with the small in the spin- tetramer, the value of is 1 in the isolated AF spin dimer given by irrespective of the spin value. It is rare that spin dimer compounds show a magnetically ordered state at atmospheric pressure and zero magnetic field. An example is the AF spin- dimer compound NHCuCl Kurniawan99 (); Matsumoto15 (); Leuenberger85 ().
We can expect an interacting AF spin- dimer model in CrVMoO from its crystal structure as shown in Fig. 1(a) Wang98 (); Knorr98 (). Only the Cr ion () has a localized spin-. The shortest distance between two Cr ions is 3.01 Å at 153 K, whereas the other Cr-Cr distances are larger than 4.96 Å Wang98 (). We found an antiferromagnetically ordered state below K. We investigated magnetism of CrVMoO using magnetization, specific heat, electron spin resonance, neutron diffraction, and inelastic neutron scattering experiments. In this paper, we report the results.
Ii Experimental and Calculation Methods
Crystalline CrVMoO powder was synthesized by a solid-state reaction. Starting materials are CrO, VO, and MoO powder. Their purity is %. A stoichiometric mixture of powder was sintered at 923 K in air for 268 h with intermediate grindings. We measured an x-ray powder diffraction pattern at room temperature using an x-ray diffractometer (RINT-TTR III, Rigaku). We confirmed that our sample was a nearly single phase of CrVMoO.
Electron spin resonance (ESR) measurements were performed using an X-band spectrometer (JES-RE3X, JEOL) at room temperature. We measured the specific heat using a physical property measurement system (Quantum Design). We measured the magnetization in magnetic fields of up to 5 T using a superconducting quantum interference device magnetometer magnetic property measurement system (Quantum Design). High-field magnetization measurements were conducted using an induction method with a multilayer pulsed field magnet installed at the Institute for Solid State Physics (ISSP), the University of Tokyo.
We carried out neutron powder diffraction experiments using the high-intensity powder diffractometer Wombat (Proposal ID P5174) at Australia’s Open Pool Australian Lightwater (OPAL) reactor in Australian Centre for Neutron Scattering in Australian Nuclear Science and Technology Organisation (ANSTO). We performed Rietveld refinements of the crystal and magnetic structures using the FULLPROF SUITE program package Rodriguez93 () with its internal tables for scattering lengths and magnetic form factors. We performed inelastic neutron scattering (INS) measurements using the inverted geometry time-of-flight spectrometer LAM-40 in High Energy Accelerator Research Organization (KEK).
We obtained the eigenenergies of isolated spin- dimers using an exact diagonalization method. We calculated the temperature dependence of the magnetic susceptibility and the magnetic-field dependence of the magnetization using the eigenenergies.
We calculated for the model shown in Fig. 1(b) using a mean-field theory based on the dimer unit (dimer mean-field theory). Finite magnetic moments were initially assumed on the Cr sites in the dimer. The mean-field Hamiltonian was then expressed by a matrix form under consideration of the external magnetic field and the molecular field from the nearest neighbor sites. The eigenstates of the mean-field Hamiltonian were used to calculate the expectation value of the ordered moments on the Cr sites. We continued this procedure until the values of the magnetic moments converged. We finally obtained a self-consistently determined solution for .
Iii Results and discussion
Figure 2 shows the derivative of the intensity of electron paramagnetic resonance (EPR) of a CrVMoO pellet at room temperature. The frequency of the incident microwave is 9.455 GHz. A clear resonance appeared. We evaluated the value to be .
Figure 3(a) shows the dependence of the specific heat of CrVMoO in zero magnetic field and the derivative of the magnetic susceptibility of CrVMoO in T. The sample was a pressed pellet and powder for and , respectively. We can see a peak around 26.5 K in and around 25.5 K in . As described later, we observed an antiferromagnetically ordered state at low in neutron powder diffraction experiments. The peak indicates the phase transition. We determined the transition temperature K mainly from the specific heat result.
The red circles in Fig. 3(b) show the dependence of of CrVMoO powder in T. The broad maximum of around 35 K indicates a low-dimensional AF spin system. The susceptibility seems to approach a finite value ( emu/mol Cr) at 0 K. The magnetic order results in the probable finite susceptibility at 0 K. The susceptibility obtained by us is close to that reported in literature Wang98 (); Botto98 ().
We considered the simple isolated AF spin- dimer model as a first approximation because of the following reasons. The spin- on Cr ions is usually a Heisenberg spin. The Cr ion is coordinated octahedrally by six oxygen ions. Symmetries of crystal fields affecting the Cr ions are nearly cubic. It is inferred that single ion anisotropy of the Cr ions is small. The green line in Fig. 3(b) shows calculated for the isolated AF spin- dimer with K and that was determined in EPR. The experimental and calculated are close to each other at high . We evaluated to be K.
An exchange interaction between two spins localized on magnetic ions with the electron configuration is dominated by an AF direct exchange interaction. Therefore, the magnitude of the exchange interaction is mainly determined by the distance between two magnetic ions. There is an empirical relation with K and Å for compounds including Cr ions () Hase14 (). The value of was calculated to be 53 K for Å. The values of and are the same in order.
The red lines in Fig. 4 show the dependence of the magnetization of CrVMoO powder measured at 1.3 and 30 K. The magnetization increases monotonically with increase in . The green lines in Fig. 4 indicate calculated for the isolated AF spin- dimer with K and . The calculated is close to the experimental at 30 K, whereas the isolated spin dimer model fails to reproduce the experimental at 1.3 K. There are and quantum magnetization plateaus in the calculated line, whereas no plateau exists in the experimental line. The and magnetization-plateau phases are polarized paramagnetic phases in which and 2, respectively. Here represents the size of the total spin of the two spins. We could not find the value of the isolated spin dimer model that reproduced the experimental at 1.3 K.
According to the results in CuInVO Hase16 (), the discrepancy between experimental and calculated is probably caused by interdimer interactions. Interdimer interactions must exist in CrVMoO to stabilize the ordered state. Interdimer interactions have a greater effect on the magnetization at lower . Therefore, the discrepancy between the experimental results and those calculated for the isolated spin dimer appears at low . We assumed the simple model shown in Fig. 1(b) as in the case of CuInVO Hase16 () and calculated using the dimer mean-field theory. The blue lines in Fig. 4 indicate calculated for the interacting spin dimer model with K, K, and . The experimental and calculated are in agreement with each other. The value is not so small compared with the value. Therefore, the antiferromagnetically ordered state appears.
We can explain qualitatively of CrVMoO. In a weakly interacting spin dimer model, magnetization plateaus exist at low . Magnetization-plateau phases are polarized paramagnetic phases without a spontaneous magnetic order. An ordered phase can appear in a magnetic-field range, where increases, between two magnetization-plateau phases. In the case of spin-, there are three types of ordered phases, phase 1, 2, and 3 at , , and , respectively. Here, and indicate magnetic fields at which the plateau starts and finishes, respectively. The phase 1 is mainly formed by and states of isolated AF spin dimers. The phase 2 is mainly formed by and states. The phase 3 is mainly formed by and states. As interdimer interactions increase, magnetic-field ranges of ordered phases are spread. When interdimer interactions are strong, the ordered phases are connected with each other. A single ordered phase is formed until the saturation of the magnetization. Therefore, the magnetization increases monotonically with increase in .
The circles in Fig. 5 show a neutron powder diffraction pattern of CrVMoO at 35 K above K. The wavelength is 2.955 Å. We performed Rietveld refinements using the space group (No. 2) to evaluate crystal structure parameters. The line on the experimental pattern indicates the result of Rietveld refinements. The line agrees with the experimental pattern. The refined crystal structure parameters are presented in Table I. The atomic positions in our results are similar to those in the literature Wang98 (); Knorr98 ().
Figure 6(a) shows neutron powder diffraction patterns of CrVMoO at 5 and 35 K. The two patterns nearly overlap each other except for around . Figure 6(b) shows the difference pattern made by subtracting the neutron powder diffraction pattern at 35 K from that at 5 K. Several magnetic reflections are apparent at 5 K. All the reflections can be indexed with the propagation vector .
The inset in Fig. 6(b) shows the dependence of the integrated intensity between 17 and including and reflections. The intensity increases with decrease in and is nearly constant below 14 K. The blue line indicates with , K, and . These values were obtained from the data above 20 K. We evaluated to be 0.26 in the spin- tetramer compound CuCdBO from the inset figure in Fig. 4 in Hase09 (). The two values of the critical exponent are close to each other. The value is 0.36, 0.33, and 1/8 for three-dimensional Heisenberg, three-dimensional Ising, and two-dimensional Ising models, respectively. In the Ising models, is smaller in the lower dimension. The spin models in CrVMoO and CuCdBO are low-dimensional AF interacting spin clusters. Therefore, the values in these compounds are smaller than that of three-dimensional Heisenberg models.
According to magnetic space groups in Litvin08 (), only a collinear magnetic structure is possible. We performed Rietveld refinements for the difference pattern using two models. Two ordered moments in each dimer are parallel in one model and antiparallel in the other one. As expected, only the antiparallel model can explain the magnetic reflections as shown in Fig. 6(b).
The magnetic structure is shown in Fig. 7 Comment01 (). An ordered moment vector is lying nearly in the plane. Its magnitude is . It is much smaller than a classical value . The GS of the spin dimer is a spin-singlet state Hase93a (); Hase93b (); Hase93c (). Therefore, the ordered moment is shrunk.
Figure 8 shows INS intensity maps of CrVMoO powder at 1.5 and 30 K. Here, and are the magnitude of the scattering vector and the energy transfer, respectively. The energy of final neutrons is 4.59 meV. We can see excitations between 2 and 7 meV at 1.5 K. The intensity of the excitations is suppressed at higher . Excitations at 1.5 K also exist below 2 meV around Å. Excitations at 30 K exist in lower energies in comparison with those at 1.5 K. The intensity is strong at low around Å.
Figure 9(a) shows the dependence of in the range of Å. The intensity at 1.5 K is the strongest around 4.5 meV. The intensity at 30 K decreases with increase in . The red circles in Fig. 9(b) show the dependence of at 1.5 K summed in the range of 4 - 5 meV. The intensity shows a peak around Å.
Considering the INS results of CuCdBO Hase15 (), we can explain qualitatively the INS results of CrVMoO using the interacting AF spin- dimer model. The blue line in Fig. 9(b) indicates the dependence of the intensity calculated for the isolated spin dimer model with the Cr-Cr distance of 3.01 Å. The experimental and calculated results are similar to each other. The first excited spin-triplet states exist at 2.2 meV (= 25 K) in the isolated AF spin- dimer. Interdimer interactions change discrete levels of excited states to excitation bands with finite widths. The excitations between 2 and 7 meV indicate the existence of the excitation bands Comment02 ().
The magnetic reflection is the strongest at . The magnetic zone center of the spin configuration shown in Fig. 7 is . The value is 0.70 Å. Therefore, the excitations at 1.5 K below 2 meV around Å are T-mode (Nambu-Goldstone mode) excitations in the vicinity of the gapless point.
The magnetic excitations are gapless below . The temperature 30 K is slightly higher than K. The bandwidths are large and the excitation gap is small. Therefore, magnetic excitations appear in low energies. Excitations from thermally excited states in the excitation bands also generate the continuous low-energy intensities at 30 K.
We could not confirm L-mode excitations because of the powder sample. We intend to make single crystals of CrVMoO and to perform INS and Raman scattering experiments on them to investigate L-mode excitations. We expect that L-mode excitations are observable because of the small ordered moment.
We investigated magnetism of CrVMoO using magnetization, specific heat, electron spin resonance, neutron diffraction, and inelastic neutron scattering experiments. An antiferromagnetically ordered state appears below K. The magnetic susceptibility of CrVMoO powder at high is close to that calculated for the isolated AF spin- dimer with the intradimer interaction value K and . We were able to explain the magnetization curves using the interacting AF spin- dimer model with the effective interdimer interaction K. We determined the magnetic structure of CrVMoO. The magnitude of ordered moments is . It is much smaller than a classical value . Two ordered moments are antiparallel in each dimer. We observed magnetic excitations in inelastic neutron scattering experiments. We can explain qualitatively the results using the interacting AF spin- dimer model. CrVMoO is a rare spin dimer compound that shows an antiferromagnetically ordered state at atmospheric pressure and zero magnetic field. Longitudinal-mode magnetic excitations may be observable in single crystalline CrVMoO.
Acknowledgements.This work was financially supported by Japan Society for the Promotion of Science (JSPS) KAKENHI (Grant No. 15K05150) and grants from National Institute for Materials Science (NIMS). M. M. was supported by JSPS KAKENHI (Grant No. 26400332). The high-field magnetization experiments were conducted under the Visiting Researcher’s Program of the Institute for Solid State Physics (ISSP), the University of Tokyo. The neutron powder diffraction experiments were performed by using the Wombat diffractometer at Australian Nuclear Science and Technology Organisation (ANSTO), Australia (proposal ID. P5174). We are grateful to S. Matsumoto for sample syntheses and x-ray diffraction measurements.
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