# Magnetic State of the Geometrically Frustrated Quasi-One-Dimensional Spin System CuMoO Studied by Thermal Conductivity

\abstWe have measured the thermal conductivity of the geometrically frustrated quasi-one-dimensional spin system CuMoO in magnetic fields. A contribution of the thermal conductivity due to spins has been observed in the thermal conductivity along the spin chains. The thermal conductivity due to phonons, , has been found to decrease by the application of a magnetic field, which has been explained as being due to the reduction in the spin gap originating from the spin-singlet dimers. Moreover, it has been found that increases with increasing field in high fields above T at low temperatures. This suggests the existence of a novel field-induced spin state and is discussed in terms of the possible spin-chirality ordering in a frustrated Mott insulator.

## 1 Introduction

The thermal conductivity in low-dimensional quantum spin systems has attracted great interest, because a large amount of thermal conductivity due to spins, namely, magnetic excitations, , has been observed along the direction where the antiferromagnetic (AF) exchange interaction is strong. In the AF spin-chain systems SrCuO[2, 3, 1] and SrCuO[1, 4] and the two-leg spin-ladder system SrCuO [5, 8, 6, 7, 9], for example, the thermal conductivity due to spinons and magnons, which are magnetic excitations in these systems, has been observed, respectively. In addition, the thermal conductivity has attracted considerable interest, because it is closely related to the magnetic state. That is, the thermal conductivity exhibits a marked change according to the change in the magnetic state, owing to the marked change in the scattering of heat carriers by magnetic excitations. In the spin-Peierls system CuGeO [10, 11] and the two-dimensional spin-dimer system SrCu(BO) [12, 13], the thermal conductivity due to phonons, , has been found to be enhanced at low temperatures below the temperature comparable to the spin-gap energy owing to the reduction in the phonon-spin scattering rate and to be suppressed by the application of a magnetic field because of the reduction in the spin gap [12, 13]. Furthermore, a marked enhancement of the thermal conductivity has been observed at low temperatures below the AF transition temperature, , in several AF spin systems [14, 15, 16, 17, 18, 19, 20]. In the frustrated spin system DyTiO, recently, the thermal conductivity has been found to be affected by the change in the state of magnetic monopoles, which are magnetic excitations in this system.[21, 22] Accordingly, the thermal conductivity is recognized as a very useful probe to detect a change in the magnetic state and a phase transition.

The compound CuMoO is a quasi-one-dimensional spin system with the quantum spin number of Cu ions. As shown in Fig. 1(a), distorted tetrahedral spin-chains composed of spin chains of Cu1 and spin dimers of Cu2 and Cu3 run along the -axis. The spin chains are arranged in the -plane as shown in Fig. 1(b). Cu spins interact with one another by AF superexchange interactions, whose magnitude has been estimated from the inelastic neutron-scattering experiment as follows [23, 24]. Both the interaction between Cu1 and Cu2, , and that between Cu1 and Cu3, , are K. The intradimer interaction between Cu2 and Cu3, , which is equal to the spin-gap energy, , of the spin dimers, is K. The intrachain interaction between Cu1’s, , is K. The interchain interaction is as negligibly weak as 2.2 K.

The magnetization and specific heat measurements have revealed that CuMoO undergoes an AF transition accompanied by weak ferromagnetism (WF) due to the Dzyaloshinsky-Moriya interaction at 8 K [25]. In the AF ordered state, the dispersion branch of magnetic excitations of the spin dimers remains together with that of the AF order [23, 24] and the direction of Cu1 spins is almost parallel to the -axis but is slightly canted from the -axis [25]. Although canted components of the magnetic moments are in disorder in zero field, they are ordered by the application of a magnetic field of 0.1 T along the -axis and of 0.8 T along the -axis. In the AF ordered state, furthermore, it has been found from dielectric constant and magnetization measurements that CuMoO shows magnetic and ferroelectric orders simultaneously without any magnetic superlattice formation,[26] which has been understood as being due to the possible charge redistribution in a frustrated Mott insulator [26, 27, 28]. The direction of the spontaneous electric polarization changes from the -axis to the -axis by the application of a magnetic field of 8 T along the -axis,[26] which has been also observed in the electron-spin-resonance spectrum of the powder sample[29]. At present, the phase diagram of CuMoO in magnetic fields at low temperatures is as shown in Fig. 2.[30, 26, 31] Nevertheless, the magnetic state of CuMoO has not yet been clarified completely. Accordingly, we have measured the thermal conductivity of single-crystal CuMoO in magnetic fields, in order to investigate the magnetic state of CuMoO as well as the existence of .

## 2 Experimental

Single crystals of CuMoO were grown by the continuous solid-state crystallization method [32]. Thermal conductivity measurements were carried out by the conventional steady-state method. One side of a rectangular single-crystal, whose typical dimensions were about 5 1 1 mm, was anchored on a heat sink of copper with indium solder. A chip-resistance of 1 k (Alpha Electronics MP1K000) was attached as a heater to the opposite side of the single crystal with GE7031 vanish. The temperature difference across the crystal (0.03–0.4 K) was measured with two Cernox thermometers (Lake Shore Cryotronics CX-1050-SD). The accuracy of the absolute value of the thermal conductivity was 10 mainly due to the uncertainty of the sample geometry. Magnetic fields up to 14 T were applied parallel to the principal crystallographic axes.

## 3 Results and Discussion

Figure 3 shows the temperature dependence of the thermal conductivity along the -, -, and -axes, , , and , of CuMoO, respectively. It is found that and perpendicular to the spin chains are similar to each other and monotonically decrease with decreasing temperature down to K. Although parallel to the spin chains also decreases with decreasing temperature down to K, on the other hand, increases with decreasing temperature from K down to . Both , and increase suddenly just below with decreasing temperature and exhibit a peak at approximately 5 K. In nonmagnetic insulators, typically increases with decreasing temperature from room temperature and shows a peak at a low temperature around 10 K. In spin-gap systems, moreover, thermal conductivity typically increases with decreasing temperature at low temperatures below the temperature comparable to the spin-gap energy. Taking into account the observation of the dispersion branch of magnetic excitations of the spin dimers,[23, 24] therefore, the monotonic decrease with decreasing temperature at high temperatures implies that the mean free path of phonons, , is strongly suppressed probably by magnetic fluctuations due to the spin frustration. The sudden increases in , , and just below are inferred to be due to the increase in owing to the marked reduction in the phonon-spin scattering rate caused by the development of the AF long-range order, as observed in several antiferromagnets [14, 15, 16, 17, 18, 19, 20].

It is found that the magnitude of is larger than those of and . Furthermore, only increases with decreasing temperature at temperatures between 20 K and , which can hardly be explained as being due to the anisotropy of . Therefore, these anisotropic behaviors of the thermal conductivity are reasonably attributed to the contribution of to , because magnetic excitations can carry heat along the -axis where the magnetic correlation is developed at low temperatures below . Such anisotropic contribution of has been observed in several low-dimensional spin systems.[2, 3, 1, 4, 5, 8, 6, 7, 9, 33, 34, 38, 19, 35, 36, 37]

Figure 4 shows the temperature dependences of and of CuMoO in magnetic fields along the -, -, and -axes, , , and , respectively. It is found that both and decrease with increasing field at low temperatures below 40 K. The decrease in by the application of a magnetic field indicates the decrease in due to the increase in the phonon-spin scattering rate, namely, the enhancement of the scattering of phonons by magnetic excitations, because the contribution of is dominant in perpendicular to the spin chains and the contribution of is negligible. It is known in spin-gap systems that is enhanced below the temperature comparable to the spin-gap energy, owing to the marked decrease in the number of magnetic excitations. Moreover, the enhancement of is suppressed by the application of a magnetic field [10, 12, 13], owing to the increase in the number of magnetic excitations because of the reduction in the spin gap. In the magnetic dispersion of CuMoO, there is a flat branch of magnetic excitations of the spin dimers [23, 24]. Such a flat magnetic branch is expected to scatter phonons strongly, because the momentum conservation law is easily satisfied in the phonon-spin scattering process. Surely, is suppressed owing to the disorder of the AF correlation induced by the application of the magnetic field in AF spin-chain systems. However, since the magnetic dispersion branch in AF spin-chain systems is dispersive, it is not easy to satisfy both the momentum and energy conservation laws in the phonon-spin scattering process. Therefore, magnetic excitations of the spin dimers are expected to scatter phonons stronger than those of the AF spin chains. Furthermore, considering that the temperature below which the suppression by the application of a magnetic field is observed is comparable to K [23, 24], the suppression of not only but also by the application of a magnetic field is interpreted as being caused by the enhancement of the phonon-spin scattering due to the reduction in the spin gap. However, neither enhancement of , , nor is observed in zero field below K comparable to . This may indicate that phonons are strongly scattered by magnetic fluctuations due to the spin frustration even at low temperatures below . Furthermore, the decrease in by the application of a magnetic field is more marked than that in . This indicates that not only but also decreases by the application of a magnetic field, because there exists the contribution of to parallel to the spin chains in zero field as described above. It is reasonable that is affected by magnetic fields up to 14 T, because K is not much larger than the energy of a magnetic field of 14 T. Namely, magnetic excitations carrying heat are scattered by the disorder of the antiferromagnetic correlation along the -axis induced by the application of a magnetic field. In fact, it has been reported that in the quasi-one-dimensional spin system SrVO with the intrachain interaction of 82 K is suppressed by the application of a magnetic field of 14 T.[36, 37] As for the behavior of the thermal conductivity in magnetic fields at low temperatures below , it is slightly complicated.

Figures 5(a)–5(d) show the magnetic-field dependences of ()/(0), and ()/(0) of CuMoO, normalized by the value in zero field, in , and at 3 and 10 K. First, we compare ()/(0) and ()/(0). It is found that both ()/(0) and ()/(0) show a complicated but similar behavior in general terms, but ()/(0) tends to decrease with increasing field more than ()/(0). Here, parallel to the spin chains is described as the sum of and , while perpendicular to the spin chains is given by only . Therefore, it is inferred that the complicated field-dependence of the thermal conductivity is due to , while monotonically decreases with increasing field, as shown in Fig. 5(e).

Next, we discuss the magnetic-field dependence of ()/(0) in order to investigate the magnetic and dielectric states, because the behavior of originating from only is expected to reflect these states through the scattering of phonons more simply than that of . It is found that the field dependence of at 3 K is very different depending on the applied-field-direction, as shown in Fig. 5(a). In low magnetic fields, increases up to T with increasing fields of and , while it decreases up to T with increasing field of . Since the long-range order of canted components of the magnetic moments in WF appears above T and T but it does not in [26], the increase in with increasing fields of and is explained as being caused by the appearance of the long-range order of the canted components in WF leading to the suppression of the phonon-spin scattering. Therefore, there is an enhanced component of in both and , as shown in Fig. 5(f). The decrease in in is explained as being caused by the increase in the phonon-spin scattering rate due to the reduction in the spin gap by the application of a magnetic field. Furthermore, it is found that starts to decrease above T with increasing fields of and , which is interpreted as being caused by both the saturation of the enhancement of by the appearance of the long-range order of the canted components in WF, as shown in Fig. 5(f), and the decrease in due to the reduction in the spin gap, as shown in Fig. 5(g).

In high magnetic fields, tends to increase above T with increasing fields of , , and , as shown in Fig. 5(a). In particular, it is remarkable that there is a kink in ()/(0) at 7.5 T, where the phase transition occurs, that is, the direction of the spontaneous electric polarization changes from the -axis to the -axis with increasing field, as shown in Fig. 2[26]. A similar kink is also observed in at T. In and , on the other hand, no anomaly suggesting any phase transitions has been observed at around 7 T in the specific heat [30] and magnetization [39] measurements. However, since the differential magnetization has shown a kink at = 6 T at 2 K, the enhancement of above T may be caused by an unknown field-induced order and/or a change in the magnetic state. Accordingly, there is an enhanced component of in , , and , as shown in Fig. 5(h). The enhancement of above T is also observed at 10 K above , as shown in Fig. 5(c).

Here, it is noted that the enhancement of above T means the increase of , because both the specific heat and velocity of phonons are usually almost independent of magnetic field. In other words, it means that the scattering rate of phonons decreases with increasing field, corresponding to the decrease in magnetic excitations and/or the development of a magnetic order. According to the calculation of the magnetic dispersion in magnetic fields by Matsumoto et al.[40], no anomaly such as any change in the ground state has been suggested at T.

Here, in order to explain the enhancement of above T, we introduce the theory proposed by Bulaevskii and Batista [27] and Khomskii [28] concerning spontaneous currents and charge redistribution in a Mott insulator regarded as a geometrically frustrated spin system. Since the ferroelectricity in CuMoO has been understood on the basis of the charge redistribution [26], the spontaneous currents may be useful to explain the enhancement of . In a geometrically frustrated Mott insulator, the exchange interaction between three spins forming a triangle causes a spontaneous current running along the triangle. This spontaneous current only appears in a non-coplanar spin-state and is proportional to the scalar spin-chirality given by , where is a spin angular momentum on the site . In the case that the spins of Cu2 and Cu3 form a spin-singlet dimer, distorted tetrahedral spin-chains can be regarded as simple spin-chains composed of only Cu1 spins and there is no chirality in the spin chain, as shown in Fig. 6(a). In the case that spin-singlet dimers are broken by the application of a magnetic field, on the other hand, finite values of spin chirality appear in the triangles, because spins revive on the Cu2 and Cu3 sites, as shown in Fig. 6(b). Therefore, it is possible that the enhancement of above T is caused by the ordering of spin chiralities, because the ordering is able to be brought about by the magnetic interaction even in the absence of any magnetically ordered state.

Finally, the magnetic-field dependence of at low temperatures below K is summarized as follows, on the basis of the scenario adopting the spin-chirality ordering. In zero field, a few excitations of spin-singlet dimers in the spin-gap state due to thermal fluctuations scatter phonons, as shown in Fig. 7(a). Since the number of magnetic excitations increases with increasing field below T by the reduction in the spin gap, is shortened because of the increase in the phonon scattering rate, as shown in Fig. 7(b). In high magnetic fields above T, the order of magnetic excitations, namely, the order of spin chiralities, is developed, as shown in Fig. 7(c), so that increases owing to the decrease in the phonon scattering rate. The reason why the enhancement of is different depending on the applied-field-direction is as follows. Spontaneous currents along the triangles composed of three spins induce orbital moments, which are coupled with the magnetic field. Therefore, the magnitude of the scalar spin-chirality might be related to the magnetic field penetrating the triangles. Accordingly, since the areas of the triangles viewed from the -axis are homogeneous, the chirality order may be homogeneous in , leading to the large enhancement of . On the other hand, since the areas of the triangles viewed from the - and -axes are inhomogeneous, the chirality order may be inhomogeneous in and , leading to the small enhancement of . To confirm this scenario adopting the spin-chirality order, further experimental and theoretical investigations are necessary.

## 4 Summary

In order to investigate the magnetic state and the existence of , we have measured , , and of CuMoO single crystals in magnetic fields up to 14 T. In zero field, it has been found that , , and are suppressed at high temperatures probably by magnetic fluctuations due to the spin frustration, while they are enhanced just below as in the case of several antiferromagnets. By the application of a magnetic field, , , and have been found to be suppressed at low temperatures below K and this has been explained as being due to the reduction in the spin gap originating from the spin-singlet dimers of Cu2 and Cu3. Since it has been found that the magnitude of parallel to the spin chains is larger than those of and and that the decrease in by the application of a magnetic field is more marked than that in , it is concluded that there exists a contribution of to . Furthermore, it has been found that the magnetic-field dependences of and at 3 and 10 K are complicated and different depending on the applied-field-direction. In low magnetic fields below T, both and have been found to decrease with increasing field due to the reduction in the spin gap. Moreover, at 3 K has been found to markedly change in and in correspondence to the appearance of the long-range order of the canted components in WF. In high magnetic fields above T, on the other hand, both and at 3 K have been found to tend to increase with increasing field. In , a kink has been observed at T in both and , owing to the field-induced phase transition. In , it has been found that the increase in above T is most marked and is observed even at 10 K above in spite of the absence of any phase transition, suggesting the existence of a novel field-induced spin state. A possible state is the ordered one of the spin chirality in a frustrated Mott insulator.

## Acknowledgment

The thermal conductivity measurements were performed at the High Field Laboratory for Superconducting Materials, Institute for Materials Research, Tohoku University. Figure 1 was drawn using VESTA [41]. This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan.

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