Abstract
When charge carriers are spatially confined to one dimension, conventional Fermiliquid theory breaks down. In such Tomonaga–Luttinger liquids, quasiparticles are replaced by distinct collective excitations of spin and charge that propagate independently with different velocities. Although evidence for spin–charge separation exists, no bulk lowenergy probe has yet been able to distinguish successfully between Tomonaga–Luttinger and Fermiliquid physics. Here we show experimentally that the ratio of the thermal and electrical Hall conductivities in the metallic phase of quasionedimensional Li_{0.9}Mo_{6}O_{17} diverges with decreasing temperature, reaching a value five orders of magnitude larger than that found in conventional metals. Both the temperature dependence and magnitude of this ratio are consistent with Tomonaga–Luttinger liquid theory. Such a dramatic manifestation of spin–charge separation in a bulk threedimensional solid offers a unique opportunity to explore how the fermionic quasiparticle picture recovers, and over what time scale, when coupling to a second or third dimension is restored.
Introduction
The success of Fermiliquid (FL) theory in describing the properties of most ordinary threedimensional metals makes it one of the triumphs of twentiethcentury theoretical physics. Its wideranging applicability is testament to the validity of describing a system of interacting electrons by mapping its lowlying quasiparticle excitations onto a Fermi gas of noninteracting electrons. Perhaps the most striking realization of this onetoone correspondence is the validity of the Wiedemann–Franz (WF) law in almost all known theoretical^{1,2,3,4} and experimental^{5,6,7} cases. The WF law states that the ratio of the electronic thermal conductivity κ_{e} to the electrical conductivity σ at a given temperature T is equal to a constant called the Lorenz number or Lorenz ratio, L_{0}=κ_{e}/σT=(π^{2}/3)(k_{B}/e)^{2} and reflects the fact that thermal and electrical currents are carried by the same fermionic quasiparticles. Although the WF law is most applicable in the zero temperature (impurity scattering) limit, it is found to hold equally well at room temperature once all inelastic scattering processes become active^{8}.
A marked deviation from the WF law is theoretically predicted when electrons are spatially confined to a single dimension. In systems that are strictly onedimensional (1D), even weak interactions destroy the single particle FL picture in favour of an exotic Tomonaga–Luttinger liquid (TLL) state in which the fundamental excitations are independent collective modes of spin and charge, referred to, respectively, as spinons and holons. As heat is transported by entropy (spin and charge) and electric current by charge alone, spin–charge separation is a viable mechanism for the violation of the WF law^{9,10,11,12}. Physically, repulsive interactions in a disordered 1D chain can inhibit the propagation of holons relative to that of spinons, leading to a strongly renormalized Lorenz number^{9}.
Experimental signatures of TLL physics have been seen in the spectral response of a number of 1D structures^{13,14,15} and bulk crystalline solids^{16,17,18,19,20}. Although the ratio κ_{e}/σT can in principle provide a direct means of distinguishing between FL and TLL states at low energies, there have been no confirmed reports to date of WF law violation in any 1D conductor. Identifying such systems, particularly bulk systems, is important as it might then allow one to tune, chemically or otherwise, the effective interchain coupling and thereby drive the system from one electronic state to the other. This would then open up the possibility of exploring the TLLtoFL crossover and the nature of the excitations in the crossover regime.
Here we report a study of the electrical and thermal conductivity tensors of the purple bronze Li_{0.9}Mo_{6}O_{17}, a quasi1D conductor whose (surfacederived) photoemission lineshapes^{19}, and density of states profiles^{20} contain features consistent with TLL theory.
Results
Electrical resistivity of Li_{0.9}Mo_{6}O_{17}
As shown in Figure 1a, Li_{0.9}Mo_{6}O_{17} possesses a set of weakly coupled zigzag chains of MoO_{6} octahedra with a holeconcentration, believed to be close to halffilling^{21}, running parallel to the crystallographic b axis. The Tdependence of the b axis resistivity, plotted in Figure 1B, varies linearly with temperature above 100 K, then as T is lowered, ρ_{b}(T) becomes superlinear. Below around 20 K, Li_{0.9}Mo_{6}O_{17} undergoes a crossover from metallic to insulatinglike behaviour, ascribed to the formation of a putative charge density wave^{22,23}. Also plotted in Figure 1b is the interchain resistivity ρ_{a}(T). The anisotropy in the electrical resistivity, ρ_{a}∼100ρ_{b} (< ρ_{c}) agrees well with optical conductivity measurements^{24} and highlights the extreme onedimensionality of the electronic system. Note that although the Tdependence of the resistivity is similar along all three crystallographic axes, the corresponding interchain mean free paths are estimated to be less than the spacing between adjacent zigzag chains, implying incoherent interchain transport, at all finite temperatures.
Thermal and electrical Hall conductivities of Li_{0.9}Mo_{6}O_{17}
In. most solids, the thermal conductivity κ can be described as the sum of two independent contributions; κ=κ_{e}+κ_{ph}, where κ_{ph} is the phonon thermal conductivity. In metals with short meanfreepaths, these two terms are comparable in magnitude, making it difficult to accurately determine the Lorenz ratio. As shown in the Methods section however, the transverse Hall conductivity κ_{xy} is purely electronic in origin (the phonon current is strictly unaffected by a magnetic field) and thus measurements of the thermal Hall effect (the thermal analogue of the electrical Hall effect) provide a means of isolating the electronic component. Moreover, the ratio of the thermal and electrical Hall conductivities κ_{xy}/σ_{xy}T, known as the Hall Lorenz number L_{xy}, is also expected to be equal to L_{0} for a Fermi liquid. In our thermal Hall apparatus (inset to Fig. 2a and described in more detail in the Methods section), a temperature gradient is applied along the conducting chain direction (b axis, taken here to be the x axis) and a magnetic field H//c (H//z) generates a Lorentz force that produces a transverse thermal gradient along a (y). The main panel in Figure 2a shows the transverse temperature difference ΔT_{y}(H), normalized by the applied power and multiplied by the sample thickness, at a number of selected temperatures. As expected, ΔT_{y} is linear and odd in field. Figure 2b shows the electrical Hall resistivity at comparable temperatures.
To determine κ_{xy} and σ_{xy} at each temperature, it is also necessary to evaluate the longitudinal terms κ_{xx} (=κ_{b}), κ_{yy} (=κ_{a}), σ_{xx} (=1/ρ_{b}) and σ_{yy} (=1/ρ_{a}) (see the Methods section for the derivation of κ_{xy}). As shown in Figure 3a, κ_{a} and κ_{b} show marked anisotropy. Due to the extreme resistivity anisotropy, κ_{a} is presumed to be purely phononic in origin and the ab anisotropy in κ(T) can be attributed either entirely to the electronic contribution within the chains or to a combination of κ_{e} and additional anisotropy in the phonon spectrum and/or scattering rate.
Figure 3b shows the resultant thermal and electrical Hall conductivities as obtained from the data presented in Figures 1b, 2a, 2b and 3a. According to the WF law, the ratio κ_{xy}/σ_{xy} should decrease (linearly) with decreasing temperature. In Li_{0.9}Mo_{6}O_{17}, however, the opposite is true; while σ_{xy} increases by a factor ∼60 between 300 K and 25 K, κ_{xy} increases by more than 3,000. This implies a change in the Hall Lorenz number L_{xy} (=κ_{xy}/σ_{xy}T) of more than 500 over the same temperature interval. The thermal Hall angle tanθ_{T}=κ_{xy}/κ_{e}, which provides a measure of the electron mobility, reaches a value ∼0.6 at T=50 K and μ_{0}H=10 Tesla comparable with that observed in elemental Cu (ref. 25). (Here we have assumed that κ_{e}=κ_{b}–κ_{a}). By comparison, the corresponding electronic Hall angle σ_{xy}/σ_{xx}∼1.25×10^{−3}, illustrating the striking difference in mobilities for entropy and charge transport in Li_{0.9}Mo_{6}O_{17}.
Lorenz ratios in Li_{0.9}Mo_{6}O_{17}
Figure 3c shows L_{xy}/L_{0}(T) for two samples over the entire temperature range studied. (Note that we have confined our Hall measurements to the metallic regime above the resistivity minimum.) At T=300 K, L_{xy}/L_{0}∼100 and as T falls, L_{xy}/L_{0} follows an inverse powerlaw∼T^{α}_{xy} (α_{xy}∼−2.3), reaching a value∼10^{5} at T=25 K. The solid triangles in Figure 3c are the corresponding estimates of L_{xx}/L_{0}=(κ_{b}–κ_{a})/σ_{b}T, that is, assuming an isotropic κ_{ph}. At T=300 K, L_{xx}/L_{0}∼7.5±1.5, rising to ∼35 at T=25 K. The horizontal dashed line is the expected FL result. Significantly, in both conventional metals like Cu (ref. 25) and Ni (ref. 26), and also in correlated metals like YBa_{2}Cu_{3}O_{6.95} (ref. 25) and URu_{2}Si_{2} (ref. 27) (whose longitudinal thermal conductivity is dominated by phonons), neither L_{xx}/L_{0} nor L_{xy}/L_{0} is ever larger than unity. (As shown in Supplementary Figure S1, we have also verified this result for Ni using our own thermal Hall apparatus). Moreover, according to standard Boltzmann transport theory^{26}, L_{xx}/L_{0}∼_{s}>/<λ_{e}> whereas L_{xy}/L_{0}∼_{s}>^{2}/<λ_{e}>^{2}, where <λ_{s(e)}> is the Fermisurface averaged mean free path for entropy (charge) transport, respectively. Hence, in ordinary metals, L_{xy}/L_{0} is expected to vary as (L_{xx}/L_{0})^{2}, as found experimentally^{25}. In Li_{0.9}Mo_{6}O_{17}, however, (L_{xx}/L_{0})^{2} does not scale with L_{xy}/L_{0} and cannot be made to scale with L_{xy}/L_{0} for any reasonable estimate for the phonon contribution to κ_{xx} (κ_{b}). This, together with the unprecedented enhancement of the WF ratio by several orders of magnitude, provides compelling evidence for the breakdown of the conventional FL picture in this quasi1D conductor.
Discussion
Before discussing the violation of the WF law in Li_{0. 9}Mo_{6}O_{17} in terms of 1D correlation physics, we first consider alternative scenarios based on localization effects. In noninteracting systems, it has been shown theoretically that the WF law is robust to impurity scattering of arbitrary strength up to the Anderson transition^{1,2,3,4}. In strongly correlated electron systems, however, the opening of a Mott gap can lead to a strong reduction of the electrical conductivity whereas the transport of heat, through spin fluctuations, can remain high^{10}. Localization corrections associated with electron–electron interactions are also believed to induce corrections to κ_{e} that do not scale with the WF ratio, leading to an enhancement in L/L_{0} (ref. 28). Such interaction corrections only appear in the diffusive limit, however, below a characteristic energy scale k_{B}T_{d}=ћ/2πτ determined by the impurity scattering rate 1/τ. Estimates for 1/τ in Li_{0.9}Mo_{6}O_{17} from inchain resistivity or magnetoresistance measurements give T_{d} values of order several tens of Kelvin. Whereas this estimate for the diffusion limit is consistent with the temperature (T_{min}∼20 K) below which the resistivity starts to increase with decreasing T, the behaviour of the magnetoresistance below T_{min} is found to be more consistent with density wave formation than localization corrections^{23}. In addition, the strong violation of the WF law is observed in the metallic regime between T_{min} and room temperature, and more significantly, above 100 K where the resistivity itself is strictly Tlinear. Collectively, these observations appear to rule out localization as the origin of the WF law violation in Li_{0.9}Mo_{6}O_{17}.
Turning now to the issue of dimensionality, the form of the enhancement of the WF ratio in Li_{0.9}Mo_{6}O_{17} is at least qualitatively consistent with the original theoretical prediction for a spinless TLL^{9}. An enhancement in L_{xx}/L_{0} originates from the fact that while heat can be transmitted through a nonmagnetic impurity (via spinons), the latter acts as a nearperfect reflector (backscatterer) of charge (holons). According to this picture, L_{xx} is predicted to be of order L_{0}/K at high T, where K is the dimensionless conductance or Luttinger parameter, and as temperature is lowered, L_{xx}/L_{0} varies as a powerlaw, L_{xx}/L_{0}∼T^{4−2/K} which diverges for K<0.5, corresponding to repulsive, longrange interactions^{9}. Although the original model considered a single impurity embedded in an interacting spinless 1D chain, the theory has been shown to apply equally to the case of multiple impurities and the inclusion of spin degrees of freedom with only minor modifications^{9,11}. Concerning the magnitude of the violation, a recent theoretical treatment of a weakly disordered TLL in the regime of large Umklapp scattering found an enhancement of L_{xx}/L_{0} of several orders of magnitude close to commensurate filling, owing to the fact that the spinon contribution to the thermal current cannot be degraded by Umklapp scattering processes^{10}. To the best of our knowledge, no other 'gapless' model can account for such a gross violation of the WF law in the metallic regime.
Although there is currently no specific theoretical framework for the thermal Hall response in a quasi1D conductor, it has been shown theoretically that a system of weakly coupled TL chains can exhibit an electrical Hall response that shows a powerlaw correction to the free Fermi (band) value in the presence of Umklapp scattering^{29}. Moreover, the divergence of L_{xy}/L_{0} with decreasing temperature (Fig. 3c) is qualitatively, if not quantitatively, similar to that found in L_{xx}/L_{0}. This divergence would seem to imply that electronic interactions in Li_{0.9}Mo_{6}O_{17} are indeed repulsive and long range, in agreement with conclusions drawn from angleresolved photoemission spectroscopy^{19} and scanning tunnelling microscopy^{20} where K, as obtained from the anomalous exponent α, is found to lie between 0.2 and 0.25. The fact that L_{xy}/L_{0} does not scale with (L_{xx}/L_{0})^{2}, as expected for a FL, might also be an intrinsic property of the TLL. Powerlaw behaviour is also predicted and indeed found in the Tdependence of the dc resistivity. Specifically, ρ_{b}(T)∼T between 100 K and 300 K (Fig. 1b). The value of K extracted from the powerlaw exponent in ρ_{b}(T) depends on the degree of commensurability. An exponent close to unity is consistent with K=0.25 for a system at or near quarterfilling^{30}. According to bandstructure calculations^{21} however, Li_{0.9}Mo_{6}O_{17} is believed to be closer to halffilling, for which a Tlinear resistivity corresponds to g∼1 (ref. 29), that is, to the noninteracting case. The origin of this inconsistency is not understood at present.
Since the TLL state is predicted to occur only in a strictly 1D interacting electron system, the inevitable coupling that exists between the chains in Li_{0.9}Mo_{6}O_{17} (to generate the Hall response) and parameterized by the interchain hopping integral t_{⊥}, ought to inhibit the observation of TLL physics in the zerofrequency limit^{31,32}. (For simplicity, we assume here that t_{⊥} is the same in both orthogonal directions). For k_{B}T<t_{⊥}, charge should hop coherently in all three dimensions, albeit with anisotropic velocities, and the system show characteristics of a Fermiliquid. Once thermal broadening is comparable with the warping of the Fermi sheets, however, hopping between chains will become incoherent, leading to a putative 3D–1D dimensional crossover. Notably, in the presence of strong interactions, the value of t_{⊥} can be significantly renormalized^{30,33} and as stated above, both band structure calculations^{21} and photoemission spectroscopy^{19} indicate that Li_{0.9}Mo_{6}O_{17} lies close to halffilling. Thus, provided nearest neighbour interactions are dominant, intrachain correlation effects in Li_{0.9}Mo_{6}O_{17} may be strong enough that its physical response is indistinguishable from that of a TLL^{33}, at least at an energy scale k_{B}T∼20 K∼2 meV. Below this scale, the FL ground state should recover, though whether or not such a crossover to FL physics can ever be realized in Li_{0.9}Mo_{6}O_{17}, given its propensity to superconducting^{22} and densitywave^{23} order, remains to be seen.
Finally, we turn to consider how one might controllably affect the interaction strength or effective dimensionality in Li_{0.9}Mo_{6}O_{17} so that the spin and charge are recombined. One obvious route to try is to vary the band filling, for example, through substitution of Mg for Li. On the basis of the diagonal principle, the ionic radius of Li^{+} ions is comparable with that of Mg^{2+}. This suggests that such substitution could proceed without adversely affecting the 1D nature of the system, allowing the role of correlations in the TLLtoFL crossover to be explored in a controlled manner. The results reported in this manuscript suggest that an accompanying study of the variation of L_{xy} with doping would provide a clear litmus test of the recovery of the FL state.
Methods
Sample growth and characterization
Single crystals of Li_{0. 9}Mo_{6}O_{17} were grown using a temperature gradient flux method^{22} and cleaved within the a–b plane. The resulting samples were faceindexed using a singlecrystal Xray diffractometer to determine the a and baxes. Thereafter barshaped crystals (approximate dimensions 700×100×20 μm^{3}) were cut from the faceindexed sample using a wiresaw.
Isolating the intrinsic inchain resistivity
To test the validity of the WF law, accurate measurements of both the electrical resistivity and thermal conductivity are essential. In a quasi1D conductor, it is especially problematic to measure the smallest of the resistivity tensor components, because even a small admixture of either of the two larger orthogonal components can give rise to erroneous values and distort the intrinsic temperature dependence of the inchain resistivity. In Li_{0.9}Mo_{6}O_{17}, reported roomtemperature values for the inchain (b axis) resistivity range from 400 μΩ cm^{23} to more than 10 mΩ cm^{34,35}. Moreover, in instances where large (that is, >1 mΩ cm) ρ_{b} values have been reported, ρ_{b} (T) tends to show a sublinear Tdependence below 300 K.
To isolate the inchain resistivity extreme care is needed to electrically short out the sample in the two directions perpendicular to the chain and thus ensure that current flow between the voltage contacts is uniaxial. In our experiments, this is achieved either by coating conductive paint or evaporating gold strips across the entire width of the sample in the two orthogonal current directions. The mounting configuration is shown as an inset in Supplementary Figure S2. The zero field measurements were carried out for 4.2 K <T<300 K in a ^{4}He dipper cryostat. In total, we measured ρ_{b}(T) of over 30 single crystals to allow us to better identify the intrinsic Tdependence of the chains. For further details of how we determined the inchain electrical resistivity, please refer to the corresponding section in the Supplementary information.
Electrical Hall effect
The electrical Hall effect was measured using the standard fourpoint method. Thin long crystals were mounted freestanding using Dupont 4929 silver epoxy. Two pairs of transverse voltage contacts were used, to check for reproducibility, and the whole process was repeated on three different samples. Owing to slight misalignment of the contacts, the measured signal contained both a symmetric (that is, magnetoresistive) and an antisymmetric part (the Hall signal). Using the relation V_{H}=½(V(B)−V(−B)), we were able to isolate the smaller antisymmetric Hall voltage V_{H}, which, when normalized by B, yields the Hall coefficient R_{H}. The uncertainty here reflects the scatter of the data from different measurements using different voltage contact pairs. To determine the Hall Lorenz number L_{xy} (=κ_{xy}/σ_{xy}T), we extract σ_{xy} from our Hall data using the following expression:
where t is the crystal thickness and I is the applied current.
Thermal conductivity measurements
For the thermal conductivity measurements, we used a zerofield setup housed in a ^{4}He flow cryostat that covers the temperature range 10 K < T < 300 K. We employed a modified steadystate method^{36}, shown schematically in Supplementary Figure S3, in which a temperature gradient, measured using a differential thermocouple, is set up across the sample through a pair of calibrated heatlinks attached to each end. The heatlinks are also differential thermocouples. The sample is suspended by the free ends of the heatlinks between two platforms that are weakly coupled to the heatbath. Each platform houses a heater that enables a temperature gradient to be set up across the sample in both directions at a fixed heatbath temperature. The thermal conductivity κ is related to the measured quantities through the relationship,
where P is the power through the crosssectional area A and l, the separation of contacts between which the temperature difference ΔT_{x} is measured.
From the temperature drop across the heatlinks, we can estimate the power entering and leaving the crystal. This is a distinct advantage of the heatlink method over other thermal conductivity experimental setups, as it ensures that any power loss due to radiative losses and heat conduction through the thermocouple wires to the heatbath is known. Although power losses due to radiation are significant (of order 20%) at high temperature, the total power loss across the sample falls to below 2% at ∼200 K. Provided the difference between the power entering the sample and leaving the sample remains below 20%, the power through the sample can be taken as the average of the input and output power.
Regarding possible extrinsic contributions to ΔT_{x}, thermocouple readings are always taken with and without heat input (in either current direction), and the direction of the heat current is reversed to remove any effect due to stray thermal gradients. Finally, data are always taken in the regime where the extracted value for the thermal conductivity (or thermal Hall conductivity) is independent of the strength of the heat current, thus eliminating nonlinear effects.
The largest errors associated with our absolute measurement of the thermal conductivity are geometrical errors associated with the finite width of the thermocouple contacts relative to their separation, as well as with the uncertainty in determining the overall dimensions of the samples. Depending on the size of the crystal to be studied, a scanning electron, focused ion beam or highpower optical microscope was used to measure its thickness. The corresponding errors are ±1 μm, ±1 μm and ±5 μm, respectively. The separation between contacts and the width of the sample was determined using a highpower optical microscope with an error of ∼7%. To measure the separation between contacts, we adopted the convention to measure the separation between the midpoint of the contacts for both the thermal and electrical measurements^{37}. From these considerations, we associate an upper bound for the error in the final values of thermal conductivity of ±15% for the samples. The reproducibility in our measurements of κ_{a} and κ_{b} is shown in the Supplementary Figures S2A and S2B respectively.
Thermal Hall measurements
By definition, the thermal Hall effect (also known as the Righi–Leduc effect) is the development of a transverse thermal gradient (ΛT_{y}) in the presence of a longitudinal thermal gradient (ΛT_{x}) and an orthogonal magnetic field (B_{z}). Crucially, ΛT_{x} is generated by a thermal heat current along x; J_{h,x}=−κ_{xx} ΛT_{x} where κ_{xx} is the total longitudinal thermal conductivity, including the phonon component. The transverse thermal current (κ_{xy} ΛT_{x}), due to the magnetic field deflection of the electronic charge carriers, is balanced by a heat current due to the 'total' thermal conductivity in the opposite direction. This is required by the condition that there is no net heat current flow in the transverse direction once steady state has been established. Again, inclusion of the relevant elements of the total thermal conductivity tensor leads to the following expression:
giving,
Hence, the thermal Hall conductivity was calculated as,
where t is the thickness of the sample.
To measure the thermal Hall effect, the same setup on which the longitudinal (zerofield) thermal conductivity measurements were performed was used, although in this case, the differential thermocouple was positioned on opposite edges of the crystal. The sample was always positioned so that the heat flow in the longitudinal direction was along the b axis of the sample. The experiment was placed in an evacuated chamber immersed in liquid helium within the coil of a superconducting magnet. Heating power was provided by a resistive element thermally connected to the experimental platform and was controlled by a Lakeshore temperature controller. During a measurement of the thermal Hall effect, the temperature was first stabilized, then a fixed field applied and finally heat current passed through the sample. The modified steady state method was then employed as before. The power loss was found to be ∼20% at room temperature, reducing to ∼10% below 200 K. The experiment is then repeated at multiple field values. To isolate the thermal Hall response, both positive and negative polarities of the magnetic field were used. As with the longitudinal thermal conductivity measurements, the heat direction was then reversed and the measurements repeated. Finally, possible extrinsic contributions to the thermal Hall conductivity, for example due to anisotropy in the thermopower, are discussed in detail in the section on possible internal heating effects in the Supplementary Information.
Additional information
How to cite this article: Wakeham, N. et al. Gross violation of the Wiedemann–Franz law in a quasionedimensional conductor. Nat. Commun. 2:396 doi: 10.1038/ncomms1406 (2011).
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Acknowledgements
We acknowledge technical and scientific assistance from A. Carrington, P.J. Heard, R.H. McKenzie, N.P. Ong and N. Shannon and collaborative support from J. He, D. Mandrus and R. Jin (ORNL). This work was supported by EPSRC (UK). N.E.H. acknowledges a Royal Society Wolfson Research Merit Award.
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N.W., A.B., X.X., JF.M. and N.E.H. designed and performed the experiments and cowrote the paper. M.G. supplied the singlecrystal samples and commented on the manuscript.
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Correspondence to Nigel E. Hussey.
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Supplementary Figures S1–S3, Supplementary Methods and Supplementary References (PDF 304 kb)
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Wakeham, N., Bangura, A., Xu, X. et al. Gross violation of the Wiedemann–Franz law in a quasionedimensional conductor. Nat Commun 2, 396 (2011). https://doi.org/10.1038/ncomms1406
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