Abstract
The notion of quantum spinliquids (QSLs), antiferromagnets with quantum fluctuationdriven disordered ground states, is now firmly established in onedimensional (1D) spin systems as well as in their ladder cousins. The spin1/2 organic insulator κ(bis(ethylenedithio)tetrathiafulvalene)_{2}Cu_{2}(CN)_{3} (κ(BEDTTTF)_{2}Cu_{2}(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 lowlying elementary spin excitations^{2,3}, 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 thermaltransport 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 measurements^{4}. The lowenergy physics of this intriguing system needs be reinterpreted in light of the present results indicating a spingapped QSL phase.
Main
In antiferromagnetically coupled spin systems, geometrical frustrations enhance quantum fluctuations. Largely triggered by the proposal of the resonatingvalencebond theory for S=1/2 degrees of freedom residing on a frustrated twodimensional (2D) triangular lattice^{5,6,7} and its possible application to highT_{c} cuprates with a doped 2D square lattice^{8,9}, realizing/detecting QSLs in 2D systems has been a longsought goal. Recently, discoveries of QSL states on S=1/2 triangular lattices have been reported in organic compounds, κ(BEDTTTF)_{2}Cu_{2}(CN)_{3} (Fig. 1, inset)^{1,10,11}, C_{2}H_{5}(CH_{3})_{3}Sb[Pd(1,3dithiole2thione4,5dithiolate)_{2}]_{2} (ref. 12) and ^{3}He thin film on graphite^{13}. In particular, the NMR spectrum of κ(BEDTTTF)_{2}Cu_{2}(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 lattice^{14}, which can be viewed as cornersharing triangles, the classical magnetically ordered state is stable in the less frustrated isotropic Heisenberg triangular lattice^{15,16}. Several ideas, such as a Hubbard model with a moderate onsite repulsion^{17}, a ring exchange model^{18} and onedimensionalization by a slight distortion from the isotropic triangular lattice^{19,20}, have been put forth to explain the absence of the longrange magnetic ordering in κ(BEDTTTF)_{2}Cu_{2}(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 lowlying excitation spectrum in the zerotemperature 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, halfinteger spin Heisenberg chains feature a massless spectrum, which enables proliferation of lowenergy spinon excitations, whereas such excitations are suppressed in the integer spin case, which has a massive spectrum^{21}.
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 lowlying excitation of κ(BEDTTTF)_{2}Cu_{2}(CN)_{3}. Very recent specificheat measurements of κ(BEDTTTF)_{2}Cu_{2}(CN)_{3} show a large linear temperaturedependent 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 spin1/2 and zero charge^{2,3}, which are to be compared with conventional (bosonic) magnons that carry spin1. However, it is premature to conclude that the QSL in κ(BEDTTTF)_{2}Cu_{2}(CN)_{3} is gapless from these measurements because the specificheat 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 required^{22}.
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 (κ_{ph}) and magnetic contributions (κ_{mag}). Indeed, a large magnetic contribution to the heat current is observed in lowdimensional spin systems^{23,24}.
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 heat^{4} and NMR relaxation rate^{1,10} around T^{*}, although no structural transition has been detected. These results obviously indicate that κ_{mag} occupies a substantial portion in κ. Various scenarios, such as a crossover to a QSL state^{4}, a phase transition associated with the pairing of spinons^{2}, spinchirality ordering^{25}, Z_{2} vortex formation^{26} and exciton condensation^{27}, have been suggested as a possible source of the anomaly at T^{*} and warrant further studies.
The thermal conductivity at μ_{0}H=0 and 10 T in the lowtemperature regime (T<300 mK) is shown in Fig. 2. A striking deviation of κ/T from a T^{2} 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 κ_{ph}/T has a T^{2} dependence, which has indeed been reported in a similar compound κ(BEDTTTF)_{2}Cu(NCS)_{2} (ref. 28). Therefore, the observed nonT^{2} dependence, together with the fact that κ is enhanced by magnetic field, definitely indicates the substantial contribution of κ_{mag} 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(BEDTTTF)_{2}Cu_{2}(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 lowlying fermionic excitations, in sharp contrast to the finite γ term reported in the heat capacity measurements^{4}. 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 excitations^{3}, which is given as
where ε_{F} 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 ε_{F}=J as in 1D spin systems^{29}, 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 system^{1}. 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 lowlying fermionic excitations, we turn to a more detailed analysis of the T dependence of κ. As seen in the inset of Fig. 4, where logκ_{mag} is plotted against logT, κ_{mag} does not show a powerlaw dependence on T. As the precise value of κ_{ph} is unknown, κ_{mag}(=κ−κ_{ph}) is estimated for several values of κ_{ph}. For each case, no linear relation is observed in this log–log plot. It should be noted that when κ_{ph} 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 κ_{ph} from the observed κ even enhances the nonpowerlaw behaviour. Thus, in spite of the ambiguity for estimating κ_{ph}, we can safely conclude that κ_{mag} at low temperatures does not exhibit a powerlaw 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 structure^{2} which will give rise to a powerlaw dependence of κ on T, in analogy to the quasiparticles in dwave 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 βvalue that implies that κ_{ph} 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 κ_{ph} (see Fig. 4). As the Arrheniustype 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 nonmagnetic excitations. On the other hand, the tiny gap value may alternatively be attributed to a proximity to a quantum critical point of Z_{2} spinliquid^{27}, 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 k_{B}T^{*}(≪J) (instead of to J itself), which may be a characteristic temperature of the QSL of κ(BEDTTTF)_{2}Cu_{2}(CN)_{3}. In any case, our lowtemperature thermaltransport 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.
Methods
κ(BEDTTTF)_{2}Cu_{2}(CN)_{3} single crystals were grown by the electrochemical method. The thermal conductivity was measured by a standard steadystate method with a oneheater–twothermometer configuration in ^{3}He 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 nondeuterated 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 temperatures^{30}. 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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Acknowledgements
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 GrantsinAid (No. 20224008 and No. 20840026) from JSPS and a GrantinAid for the Global COE Program ‘The Next Generation of Physics, Spun from Universality and Emergence’ from MEXT of Japan.
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M.Y., N.N., Y.K., T. Shibauchi and Y.M. carried out measurements, data analysis and discussion. T. Sasaki, N.Y. and N.K. prepared the samples. S.F. gave theoretical advice.
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Yamashita, M., Nakata, N., Kasahara, Y. et al. Thermaltransport measurements in a quantum spinliquid state of the frustrated triangular magnet κ(BEDTTTF)_{2}Cu_{2}(CN)_{3}. Nature Phys 5, 44–47 (2009). https://doi.org/10.1038/nphys1134
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