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Gross Violation of the Wiedemann&Ndash;Franz ARTICLE Received 25 Feb 2011 | Accepted 20 Jun 2011 | Published 19 Jul 2011 DOI: 10.1038/ncomms1406 Gross violation of the Wiedemann – Franz law in a quasi-one-dimensional conductor Nicholas Wakeham1 , Alimamy F. Bangura1 , 2 , Xiaofeng Xu1 , 3 , Jean-Francois Mercure1 , Martha Greenblatt 4 & N i g e l E . H u s s e y1 When charge carriers are spatially confi ned to one dimension, conventional Fermi-liquid 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 low-energy probe has yet been able to distinguish successfully between Tomonaga – Luttinger and Fermi- liquid physics. Here we show experimentally that the ratio of the thermal and electrical Hall conductivities in the metallic phase of quasi-one-dimensional Li0.9 Mo6 O17 diverges with decreasing temperature, reaching a value fi ve 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 three-dimensional 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. 1 H. H. Wills Physics Laboratory, University of Bristol , Tyndall Avenue, Bristol BS8 1TL , UK . 2 RIKEN (The Institute of Physical and Chemical Research) , Wako , Saitama 351-0198 , Japan . 3 Department of Physics, Hangzhou Normal University , Hangzhou 310036 , China . 4 Department of Chemistry and Chemical Biology, Rutgers University , Piscataway , New Jersey 08854 , USA . Correspondence and requests for materials should be addressed to N.E.H. (email: [email protected] ). NATURE COMMUNICATIONS | 2:396 | DOI: 10.1038/ncomms1406 | www.nature.com/naturecommunications 1 © 2011 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms1406 κ κ κ κ he success of Fermi-liquid (FL) theory in describing the sum of two independent contributions; = e + ph , where ph is the properties of most ordinary three-dimensional metals makes phonon thermal conductivity. In metals with short mean-free-paths, T it one of the triumphs of twentieth-century theoretical these two terms are comparable in magnitude, making it diffi cult physics. Its wide-ranging applicability is testament to the validity to accurately determine the Lorenz ratio. As shown in the Methods κ of describing a system of interacting electrons by mapping its low- section however, the transverse Hall conductivity xy is purely elec- lying quasiparticle excitations onto a Fermi gas of non-interacting tronic in origin (the phonon current is strictly unaff ected by a magnetic electrons. Perhaps the most striking realization of this one-to-one fi eld) and thus measurements of the thermal Hall eff ect (the thermal correspondence is the validity of the Wiedemann – Franz (WF) law analogue of the electrical Hall eff ect) provide a means of isolating the in almost all known theoretical 1 – 4 and experimental5 – 7 cases. Th e electronic component. Moreover, the ratio of the thermal and electri- κ σ WF law states that the ratio of the electronic thermal conductivity cal Hall conductivities xy / xy T , known as the Hall Lorenz number κ σ e to the electrical conductivity at a given temperature T is equal L xy , is also expected to be equal to L 0 for a Fermi liquid. In our thermal κ σ to a constant called the Lorenz number or Lorenz ratio, L0 = e / T = Hall apparatus (inset to Fig. 2a and described in more detail in π2 2 ( / 3)( kB / e ) and refl ects the fact that thermal and electrical currents the Methods section), a temperature gradient is applied along the are carried by the same fermionic quasiparticles. Although the WF conducting chain direction ( b axis, taken here to be the x axis) and a law is most applicable in the zero temperature (impurity scattering) magnetic fi eld H / / c ( H / / z ) generates a Lorentz force that produces a limit, it is found to hold equally well at room temperature once all transverse thermal gradient along a ( y ). Th e main panel in Figure 2a 8 Δ inelastic scattering processes become active . shows the transverse temperature diff erence T y ( H ), normalized by A marked deviation from the WF law is theoretically predicted the applied power and multiplied by the sample thickness, at a Δ when electrons are spatially confi ned to a single dimension. In number of selected temperatures. As expected, T y is linear and odd systems that are strictly one-dimensional (1D), even weak interac- in fi eld. Figure 2b shows the electrical Hall resistivity at comparable tions destroy the single particle FL picture in favour of an exotic temperatures. Tomonaga – Luttinger liquid (TLL) state in which the fundamental excitations are independent collective modes of spin and charge, a referred to, respectively, as spinons and holons. As heat is trans- ported by entropy (spin and charge) and electric current by charge alone, spin – charge separation is a viable mechanism for the viola- tion of the WF law 9 – 12 . Physically, repulsive interactions in a disor- dered 1D chain can inhibit the propagation of holons relative to that MoO6 of spinons, leading to a strongly renormalized Lorenz number 9 . MoO Experimental signatures of TLL physics have been seen in the 4 spectral response of a number of 1D structures 13 – 15 and bulk crys- Li 16 – 20 κ σ talline solids . Although the ratio e / T can in principle pro- a vide a direct means of distinguishing between FL and TLL states at low energies, there have been no confi rmed reports to date of c WF law violation in any 1D conductor. Identifying such systems, b particularly bulk systems, is important as it might then allow one to tune, chemically or otherwise, the eff ective interchain coupling b and thereby drive the system from one electronic state to the other. 350 Th is would then open up the possibility of exploring the TLL-to-FL crossover and the nature of the excitations in the crossover regime. 300 Here we report a study of the electrical and thermal conductiv- 250 ity tensors of the purple bronze Li0.9 Mo6 O17 , a quasi-1D conductor 19 whose (surface-derived) photoemission lineshapes , and density of cm) a/100 states profi les 20 contain features consistent with TLL theory. 200 μΩ ( 150 Results Electrical resistivity of Li0.9 Mo6 O17 . As shown in Figure 1a , 100 Li Mo O possesses a set of weakly coupled zigzag chains of 0.9 6 17 b MoO6 octahedra with a hole-concentration, believed to be close to 50 half-fi lling 21 , running parallel to the crystallographic b axis. Th e T - dependence of the b axis resistivity, plotted in Figure 1B , varies linearly 0 ρ 0 50 100 150 200 250 300 with temperature above 100 K, then as T is lowered, b ( T ) becomes T (K) superlinear. Below around 20 K, Li0.9 Mo6 O17 undergoes a crossover from metallic to insulating-like behaviour, ascribed to the formation of a putative charge density wave 22,23 . Also plotted in Figure 1b Figure 1 | Crystal structure and electrical resistivity of the quasi-one- ρ is the interchain resistivity a ( T ). Th e anisotropy in the electrical dimensional metal Li0.9 Mo6 O17 . ( a ) Three-dimensional crystal structure showing ρ ρ ρ resistivity, a ~ 100 b ( c ) agrees well with optical conductivity isolated, conducting, zigzag chains of MoO6 octahedra along the b axis in dark 24 measurements and highlights the extreme one-dimensionality purple and non-conducting octahedra and MoO4 tetrahedra in light pink. Li ions ρ ρ of the electronic system. Note that although the T -dependence are shown as green spheres. ( b ) Intra-chain ( b ) and inter-chain ( a ) resistivities ρ of the resistivity is similar along all three crystallographic axes, of Li0.9 Mo6 O17 as a function of temperature. Note that a has been scaled by ρ ρ ± the corresponding interchain mean free paths are estimated to be a factor of 1 / 100. The error bars in a and b ( 7 % and 12 % , respectively) less than the spacing between adjacent zigzag chains, implying represent the uncertainty in the determination of the sample dimensions and the incoherent interchain transport, at all fi nite temperatures. standard deviation of the room temperature resistivity of the subset of curves 1 – 5 shown with solid lines in Supplementary Figure S1a, respectively. For details ρ Th ermal and electrical Hall conductivities of Li 0.9 Mo6 O17 . I n . of how b was determined, please refer to the section on isolating the intrinsic most solids, the thermal conductivity κ can be described as the in-chain resistivity in the Supplementary information. 2 NATURE COMMUNICATIONS | 2:396 | DOI: 10.1038/ncomms1406 | www.nature.com/naturecommunications © 2011 Macmillan Publishers Limited. All rights reserved. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms1406 ARTICLE ab0.02 10 110 K 100 K 0.015 150 K 210 K 0.01 5 150 K 300 K 0.005 200 K ) xy –1 300 K (10 0 0 –8 Ω (km W m) P / Tt Δ –0.005 Heat Heat pipe pipe Heater B Heater –0.01 –5 –0.015 –0.02 –10 –10 –50 5 10 –10–5 0 5 10 0H (T) 0H (T) Δ Figure 2 | Thermal and electrical Hall measurements of Li0.9 Mo6 O17 . ( a ) Transverse temperature difference T y per unit power (multiplied by the sample thickness) as a function of applied magnetic fi eld at various temperatures.
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