The notion of quantum spin-liquids (QSLs), antiferromagnets with quantum fluctuation-driven disordered ground states, is now firmly established in one-dimensional (1D) spin systems as well as in their ladder cousins. The spin-1/2 organic insulator κ-(bis(ethylenedithio)-tetrathiafulvalene)2Cu2(CN)3 (κ-(BEDT-TTF)2Cu2(CN)3; ref. 1) with a 2D triangular lattice structure is very likely to be the first experimental realization of this exotic state in D≥2. Of crucial importance is to unveil the nature of the low-lying elementary spin excitations2,3, particularly the presence/absence of a ‘spin gap’, which will provide vital information on the universality class of this putative QSL. Here, we report on our thermal-transport measurements carried out down to 80 mK. We find, rather unexpectedly, unambiguous evidence for the absence of a gapless excitation, which sharply contradicts recent reports of heat capacity measurements4. The low-energy physics of this intriguing system needs be reinterpreted in light of the present results indicating a spin-gapped QSL phase.
In antiferromagnetically coupled spin systems, geometrical frustrations enhance quantum fluctuations. Largely triggered by the proposal of the resonating-valence-bond theory for S=1/2 degrees of freedom residing on a frustrated two-dimensional (2D) triangular lattice5,6,7 and its possible application to high-Tc cuprates with a doped 2D square lattice8,9, realizing/detecting QSLs in 2D systems has been a long-sought goal. Recently, discoveries of QSL states on S=1/2 triangular lattices have been reported in organic compounds, κ-(BEDT-TTF)2Cu2(CN)3 (Fig. 1, inset)1,10,11, C2H5(CH3)3Sb[Pd(1,3-dithiole-2-thione-4,5-dithiolate)2]2 (ref. 12) and 3He thin film on graphite13. In particular, the NMR spectrum of κ-(BEDT-TTF)2Cu2(CN)3 exhibits no signs of magnetic ordering down to ∼30 mK, which is some four orders of magnitude below the exchange coupling J∼250 K (refs 1, 11). These findings aroused great interest because it is generally believed that whereas a QSL state is realized in the strongly frustrated S=1/2 2D kagome lattice14, which can be viewed as corner-sharing triangles, the classical magnetically ordered state is stable in the less frustrated isotropic Heisenberg triangular lattice15,16. Several ideas, such as a Hubbard model with a moderate onsite repulsion17, a ring exchange model18 and one-dimensionalization by a slight distortion from the isotropic triangular lattice19,20, have been put forth to explain the absence of the long-range magnetic ordering in κ-(BEDT-TTF)2Cu2(CN)3. Nevertheless, the origin for the QSL state remains unresolved.
To understand the nature of novel QSL states, knowledge on the structure of the low-lying excitation spectrum in the zero-temperature limit, particularly the absence/presence of a spin gap, is indispensable, bearing immediate implications on the spin correlations of the ground state, as well as on the quantum numbers carried by each elementary excitation. For instance in 1D, half-integer spin Heisenberg chains feature a massless spectrum, which enables proliferation of low-energy spinon excitations, whereas such excitations are suppressed in the integer spin case, which has a massive spectrum21.
As it is not possible to directly probe the microscopic spin structure using neutron scattering owing to the compound’s organic nature, thermodynamic measurements must be adopted to unveil the low-lying excitation of κ-(BEDT-TTF)2Cu2(CN)3. Very recent specific-heat measurements of κ-(BEDT-TTF)2Cu2(CN)3 show a large linear temperature-dependent contribution, γ∼15 mJ K−2 mol−1 (ref. 4), which suggests the presence of gapless excitations, similar to the electronic specific heat in metals. This observation provides strong support for several theoretical models, including a QSL with gapless ‘spinons’, which, like its 1D predecessors are (fermionic) elementary excitations that carry spin-1/2 and zero charge2,3, which are to be compared with conventional (bosonic) magnons that carry spin-1. However, it is premature to conclude that the QSL in κ-(BEDT-TTF)2Cu2(CN)3 is gapless from these measurements because the specific-heat data are plagued by a very large nuclear Schottky contribution below 1 K (ref. 4), which would necessarily lead to ambiguity. Incorporation of a probe that is free from such a contamination is strongly required22.
As pointed out in ref. 3, thermal conductivity (κ) measurements are highly advantageous as probes of elementary excitations in QSLs, because κ is sensitive exclusively to itinerant excitations and is totally insensitive to localized entities such as are responsible for Schottky contributions. The heat is carried primarily by acoustic phonons (κph) and magnetic contributions (κmag). Indeed, a large magnetic contribution to the heat current is observed in low-dimensional spin systems23,24.
As shown in Fig. 1, the thermal conductivity exhibits an unusual behaviour characterized by a hump structure around T*≃6 K. A similar hump is observed in the magnetic part of the specific heat4 and NMR relaxation rate1,10 around T*, although no structural transition has been detected. These results obviously indicate that κmag occupies a substantial portion in κ. Various scenarios, such as a crossover to a QSL state4, a phase transition associated with the pairing of spinons2, spin-chirality ordering25, Z2 vortex formation26 and exciton condensation27, have been suggested as a possible source of the anomaly at T* and warrant further studies.
The thermal conductivity at μ0H=0 and 10 T in the low-temperature regime (T<300 mK) is shown in Fig. 2. A striking deviation of κ/T from a T2 dependence is observed for both samples; both curves exhibit a convex trend. At such low temperatures, the mean free path of phonons is as long as the crystal size and κph/T has a T2 dependence, which has indeed been reported in a similar compound κ-(BEDT-TTF)2Cu(NCS)2 (ref. 28). Therefore, the observed non-T2 dependence, together with the fact that κ is enhanced by magnetic field, definitely indicates the substantial contribution of κmag in κ even in this T range.
The results shown in Fig. 2 provide key information on the elementary excitations from the QSL state of κ-(BEDT-TTF)2Cu2(CN)3. Most importantly, it is extremely improbable from the experimental data that κ/T in the T→0 K regime has a finite residual value for data of both samples in zero field and that of sample A under 10 T. (Indeed, a simple extrapolation of both data in zero field even gives a negative intersect.) These results lead us to conclude that κ/T vanishes at T=0 K. It should be stressed that the vanishing κ/T immediately indicates the absence of low-lying fermionic excitations, in sharp contrast to the finite γ term reported in the heat capacity measurements4. We believe that the heat capacity measurements incorrectly suggest the presence of gapless excitation, possibly owing to the large Schottky contribution at low temperatures.
The present conclusion is reinforced by comparing the data with the thermal conductivity calculated by assuming a spinon Fermi surface with gapless excitations3, which is given as
where εF is the Fermi energy, m is the electron mass, A is the unit cell area of the layer, d is the interlayer distance and τ is the impurity scattering time. Estimating εF=J as in 1D spin systems29, we compare our result with equation (1) as shown in Fig. 3. It is evident that equation (1) yields κ/T that increases with decreasing T for both clean and dirty cases and is opposite to the observation. Moreover, to obtain the same magnitude of κ/T in this model at the lowest temperature, we need to assume that the mean free path is only a few times longer than the lattice constant a. However, such a large concentration of the impurity is highly unlikely in this clean system1. Thus, the theory based on a gapless fermionic spinon picture is incompatible with the present results, although it may be applicable to other systems.
Having established the absence of the low-lying fermionic excitations, we turn to a more detailed analysis of the T dependence of κ. As seen in the inset of Fig. 4, where logκmag is plotted against logT, κmag does not show a power-law dependence on T. As the precise value of κph is unknown, κmag(=κ−κph) is estimated for several values of κph. For each case, no linear relation is observed in this log–log plot. It should be noted that when κph is increased, the nonlinearity becomes more pronounced. This is also manifested by the index n=(dlogκ/dlogT) plotted in the same figure where n increases steeply with decreasing temperature, and subtracting κph from the observed κ even enhances the non-power-law behaviour. Thus, in spite of the ambiguity for estimating κph, we can safely conclude that κmag at low temperatures does not exhibit a power-law temperature dependence. These results place further constraints on the theoretical description of the excitation spectrum; for example, the nodal excitations that may be expected in systems with an anisotropic gap structure2 which will give rise to a power-law dependence of κ on T, in analogy to the quasiparticles in d-wave superconductors, are also absent.
The absence of the gapless excitation implies the presence of a spin gap in the excitation spectrum. To estimate the magnitude of the spin gap, we try to fit the data to
as shown in an Arrhenius plot in Fig. 4. The best fit for the 0 (10) T data gives Δ=0.46 (0.38) K and a β-value that implies that κph is roughly 1/4 of the total κ at 100 mK. We note that the amplitude of Δ is little affected by the choice of κph (see Fig. 4). As the Arrhenius-type behaviour is observed in only one order range of κ, the estimation of the gap size may have a large ambiguity. Nevertheless, we can safely conclude that the estimated gap value from Fig. 4 is strikingly small compared with J (Δ∼J/500) and insensitive to magnetic fields.
This field insensitivity is consistent with a theory of a gapped QSL (ref. 7) with a finite energy gap for both magnetic and non-magnetic excitations. On the other hand, the tiny gap value may alternatively be attributed to a proximity to a quantum critical point of Z2 spin-liquid27, or as a result of a slight anisotropy of J (ref. 19). However, at present, the origin of the spin gap is an open question. It is tempting to associate the extremely small gap value with kBT*(≪J) (instead of to J itself), which may be a characteristic temperature of the QSL of κ-(BEDT-TTF)2Cu2(CN)3. In any case, our low-temperature thermal-transport measurements demonstrate that the fermionic spinons, if present, will experience an instability in this system, which will generate a small gap in the spin excitation spectrum.
κ-(BEDT-TTF)2Cu2(CN)3 single crystals were grown by the electrochemical method. The thermal conductivity was measured by a standard steady-state method with a one-heater–two-thermometer configuration in 3He and dilution refrigerators. The thermal current was applied within the 2D plane and the magnetic field was applied perpendicular to the plane. We have measured several deuterated and non-deuterated crystals and observed no significant sample dependence. It has been reported that in superconductors, thermal decoupling between the electron and phonon conductions can be caused by the poor contacts at very low temperatures30. It could be argued that such a decoupling may occur in the phonons and the spinons, and may lead to apparent absence of finite κ/T at T→0 K. However, we note that this is inconsistent with the observed increase of κ with H (shown in Supplementary Information). Because the magnetic field decreases the number of spinons, κ should decrease with H owing to the further reduction of the coupling. Moreover, we measured the thermal conductivity on the samples with the contact resistance ranging from 1 to 20 Ω and found no serious difference in the thermal conductivity at low temperatures.
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We would like to thank G. Baskaran, L. N. Bulaevskii, Y. Hayashi, N. Kawakami, H. Kawamura, Y. Nakazawa, S. Sachdev, A. Tanaka and S. Watanabe for valuable discussion. This work was supported by Grants-in-Aid (No. 20224008 and No. 20840026) from JSPS and a Grant-in-Aid for the Global COE Program ‘The Next Generation of Physics, Spun from Universality and Emergence’ from MEXT of Japan.
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Yamashita, M., Nakata, N., Kasahara, Y. et al. Thermal-transport measurements in a quantum spin-liquid state of the frustrated triangular magnet κ-(BEDT-TTF)2Cu2(CN)3. Nature Phys 5, 44–47 (2009). https://doi.org/10.1038/nphys1134
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