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A disorder-enhanced quasi-one-dimensional superconductor

A disorder-enhanced quasi-one-dimensional superconductor ARTICLE Received 26 Oct 2015 | Accepted 17 Jun 2016 | Published 22 Jul 2016 DOI: 10.1038/ncomms12262 OPEN A disorder-enhanced quasi-one-dimensional superconductor 1, 1, 2 3 4,5 4 4 A.P. Petrovic´ *, D. Ansermet *, D. Chernyshov , M. Hoesch , D. Salloum , P. Gougeon , M. Potel , 6 1 L. Boeri & C. Panagopoulos A powerful approach to analysing quantum systems with dimensionality d41 involves adding a weak coupling to an array of one-dimensional (1D) chains. The resultant quasi-1D (q1D) systems can exhibit long-range order at low temperature, but are heavily influenced by interactions and disorder due to their large anisotropies. Real q1D materials are therefore ideal candidates not only to provoke, test and refine theories of strongly correlated matter, but also to search for unusual emergent electronic phases. Here we report the unprecedented enhancement of a superconducting instability by disorder in single crystals of Na Mo Se , 2  d 6 6 a q1D superconductor comprising MoSe chains weakly coupled by Na atoms. We argue that disorder-enhanced Coulomb pair-breaking (which usually destroys superconductivity) may be averted due to a screened long-range Coulomb repulsion intrinsic to disordered q1D materials. Our results illustrate the capability of disorder to tune and induce new correlated electron physics in low-dimensional materials. Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore. 2 3 Swiss-Norwegian Beamlines, European Synchrotron Radiation Facility, 6 rue Jules Horowitz, F-38043 Grenoble Cedex, France. Diamond Light Source, Harwell Campus, Didcot OX11 0DE, Oxfordshire, UK. Sciences Chimiques, CSM UMR CNRS 6226, Universite´ de Rennes 1, Avenue du Ge´ne´ral Leclerc, 5 6 35042 Rennes Cedex, France. Faculty of Science III, Lebanese University, PO Box 826, Kobbeh-Tripoli, Lebanon. Institute for Theoretical and Computational Physics, TU Graz, Petersgasse 16, 8010 Graz, Austria. * These authors contributed equally to this work. Correspondence and requests for materials should be addressed to A.P.P. (email: appetrovic@ntu.edu.sg) or to C.P. (email: christos@ntu.edu.sg). NATURE COMMUNICATIONS | 7:12262 | DOI: 10.1038/ncomms12262 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12262 eakly-interacting electrons in a three-dimensional high polarizability of disordered q1D materials. The combination (3D) periodic potential are well-described by of disorder and q1D crystal symmetry constitutes a new recipe WLandau–Fermi liquid theory , in which the free for strongly correlated electron liquids with tunable electronic electrons of a Fermi gas become dressed quasiparticles with properties. renormalized dynamical properties. Conversely, in the one- dimensional (1D) limit a Tomonaga–Luttinger liquid (TLL) is Results 2,3 formed , where single-particle excitations are replaced by highly Crystal and electronic structure of Na Mo Se .Na Mo Se 2  d 6 6 2  d 6 6 correlated collective excitations. So far, it has proved difficult to belongs to the q1D M Mo Se family (M ¼ Group IA alkali 2 6 6 interpolate theoretically between these two regimes, either by metals, Tl, In) which crystallize with hexagonal space group strengthening electron–electron (e –e ) interactions in 3D, or P6 /m. The structure can be considered as a linear condensation 4,5 by incorporating weak transverse coupling into 1D models . The of Mo Se clusters into infinite-length (Mo Se ) chains parallel 6 8 6 6 N invariable presence of disorder in real materials places further to the hexagonal c-axis, weakly coupled by M atoms (Fig. 1a). The demands on theory, particularly in the description of ordered q1D nature of these materials is apparent from the needle-like electronic ground states. Q1D systems such as nanowire ropes, morphology of as-grown crystals (Fig. 1b; see Methods for growth filamentary networks or single crystals with uniaxial anisotropy details). Ab initio calculations (Supplementary Note I) using therefore represent an opportunity to experimentally probe what density functional theory reveal an electronic structure which is theories aspire to model: strongly correlated electrons subject to uniquely simple among q1D metals. A single spin-degenerate disorder in a highly anisotropic 3D environment. band of predominant Mo d character crosses the Fermi energy xz Physical properties of q1D materials may vary considerably E at half-filling (Fig. 1c, Supplementary Fig. 1), creating a 1D with temperature. TLL theory is expected to be valid at Fermi surface composed of two sheets lying close to the Brillouin elevated temperatures, since electrons cannot hop coherently zone boundaries at p/c (where c is the c-axis lattice parameter). perpendicular to the high-symmetry axis and q1D systems behave The warping of these sheets (and hence the coupling between as decoupled arrays of 1D filaments. Phase-coherent single- (Mo Se ) chains) is controlled by the M cation, yielding 6 6 N particle hopping can only occur below temperature T rt x > values for t ranging from 230 K (M ¼ Tl) to 30 K (M ¼ Rb) (where t is the transverse hopping integral), at which a (Supplementary Fig. 2). In addition to tuning the dimensionality, dimensional crossover to an anisotropic quasi-3D (q3D) electron M also controls the ground state: M ¼ Tl, In are 4,6 liquid is anticipated . The properties of such q3D liquids remain 15,16 superconductors , while M ¼ K, Rb become insulating at low largely unknown, especially the role of electronic correlations in 16,17 temperature . determining the ground state. At low temperature, a TLL is Within the M Mo Se family, M ¼ Na is attractive for two 2 6 6 unstable to either density wave (DW) or superconducting reasons. First, we calculate an intermediate t ¼ 120 K, suggesting fluctuations, depending on whether the e –e interaction that Na Mo Se lies at the threshold between superconducting 2  d 6 6 is repulsive (due to Coulomb forces) or attractive (from and insulating instabilities. Second, the combination of the small electron–phonon coupling). Following dimensional crossover, Na cation size and a high growth temperature (1750 C) results in the influence of such interactions in the q3D state is unclear. substantial Na vacancy formation during crystal synthesis. Since As an example, electrical transport in the TLL state of the Na atoms are a charge reservoir for the (Mo Se ) chains, 6 6 N the q1D purple bronze Li Mo O is dominated by repulsive 0.9 6 17 these vacancies will reduce E and lead to an incommensurate 7,8 e –e interactions , yet a superconducting transition occurs band filling. Despite the reduction in carrier density, the density for temperatures below 1.9 K. of states N(E ) remains constant for Na (Fig. 1d, F 1.5-2.1 Disorder adds further complication to q1D materials due Supplementary Note I). Energy-dispersive X-ray (EDX) spectro- to its tendency to localize electrons at low temperature. For metry on our crystals indicates Na contents from 1.7 to 2, dimensionality dr2, localization occurs for any non-zero comfortably within this range. This is confirmed by synchrotron disorder; in contrast, for d42 a critical disorder is required and X-ray diffraction (XRD) on three randomly-chosen crystals: a mobility edge separates extended from localized states. ± ± structural refinements reveal Na deficiencies of 11 1%, 11 2% The question of whether a mobility edge can form in q1D and 13 4% (that is, d ¼ 0.22, 0.22, 0.26), but the (Mo Se ) 6 6 N materials after crossover to a q3D liquid state is open, as is the chains remain highly ordered. No deviation from the M Mo Se 2 6 6 microscopic nature of the localized phase. Disorder also structure is observed between 293 and 20 K, ruling out any lattice renormalizes e –e interactions, leading to a dynamic ampli- distortions such as the Peierls transition, which often afflicts q1D fication of the Coulomb repulsion and a weaker enhancement of metals. To probe the Na vacancy distribution, we perform diffuse 10–13 phonon-mediated e –e attraction, that is, Cooper pairing . X-ray scattering experiments on the d ¼ 0.26 crystal. No trace of We therefore anticipate that disorder should strongly suppress any Huang scattering (from clustered Na vacancies) or structured superconductivity in q1D materials, unless the Coulomb diffuse scattering from short-range vacancy ordering is observed interaction is unusually weak or screened. (Supplementary Fig. 3, Supplementary Note II). Na vacancies In this work, we show that the q1D superconductor therefore create an intrinsic, random disorder potential in Na Mo Se provides a unique environment in which to study 2  d 6 6 Na Mo Se single crystals. 2  d 6 6 the interplay between dimensionality, electronic correlations and disorder. Although Na Mo Se is metallic at room tempera- 2  d 6 6 ture, the presence of Na vacancy disorder leads to electron Normal-state electrical transport. We first examine the electrical localization and a divergent resistivity r(T) at low temperature, transport at high energy for signatures of disorder and one- prior to a superconducting transition. In contrast with all other dimensionality. The temperature dependence of the resistivity known superconductors, the onset temperature for super- r(T) for six randomly-selected Na Mo Se crystals A–F is 2  d 6 6 conducting fluctuations T is positively correlated with the level shown in Fig. 2a. r(300 K) increases by 41 order of magnitude pk of disorder. Normal-state electrical transport measurements also from crystal A to F (Fig. 2b): such large differences between display signatures of an attractive e –e interaction, which is crystals cannot be attributed to changes in the carrier density due consistent with disorder-enhanced superconductivity. A plausible to Na stoichiometry variation and must instead arise from explanation for these phenomena is an intrinsic screening of the disorder. Despite the variance in r(300 K), the evolution of r(T) long-range Coulomb repulsion in Na Mo Se , arising from the is qualitatively similar in all crystals. On cooling, r(T) exhibits 2  d 6 6 2 NATURE COMMUNICATIONS | 7:12262 | DOI: 10.1038/ncomms12262 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12262 ARTICLE a b Na Mo Se c d 0.0 E –0.4 –0.8 –1.2 Γ MK Γ AL H A –2 –1 0 E–E (eV) Figure 1 | Quasi-one-dimensional crystal and electronic structures in Na Mo Se . (a) Hexagonal crystal structure of Na Mo Se , viewed 2  d 6 6 2  d 6 6 perpendicular and parallel to the c-axis. From synchrotron X-ray diffraction experiments, we measure the a- and c-axis lattice parameters to be 8.65 Å and 4.49 Å, respectively at 293 K (Supplementary Note II). (b) Electron micrograph of a typical Na Mo Se crystal. Scale bar, 300mm. (c) Calculated 2  d 6 6 energy-momentum dispersion of the conduction band within the hexagonal Brillouin zone, highlighting the large bandwidth and minimal dispersion perpendicular to the chain axis. (d) Electronic density of states N(E) around the Fermi level in Na Mo Se . 2 6 6 metallic behaviour before passing through a broad minimum at increasing disorder) and diverging at lower temperature. These T and diverging at lower temperature. T falls from 150 K to features are consistently reproduced in our data. min min B70 K as r(300 K) decreases (Fig. 2c), suggesting that the Within the disordered TLL paradigm, our high-temperature divergence in r(T) and the disorder level are linked. transport data indicate that the e –e interaction is Upturns or divergence in r(T) have been widely reported in attractive, that is, K 41. This implies that electron–phonon 18–22 q1D materials and variously attributed to localization , coupling dominates over Coulomb repulsion and suggests that 23 24,25 multiband TLL physics , DW formation , incipient density the Coulomb interaction may be intrinsically screened in 16 8 fluctuations and proximity to Mott instabilities . Differentiating Na Mo Se . A quantitative analysis of the low-temperature 2  d 6 6 between these mechanisms has proved challenging, in part due to divergence in r(T) provides further support for the influence of the microscopic similarity between localized electrons and disorder, as well as a weak/screened Coulomb repulsion. We have randomly-pinned DWs in 1D. We briefly remark that the attempted to fit r(T) using a wide variety of resistive mechanisms: broad minimum in r(T)inNa Mo Se contrasts strongly with gap formation (Arrhenius activation), repulsive TLL power 2  d 6 6 the abrupt jumps in r(T) for nesting-driven DW materials such laws, weak and strong localization (Supplementary Fig. 4, as NbSe (ref. 26), while any Mott transition will be suppressed Supplementary Note III). Among these models, only Mott due to the non-stoichiometric Na content. variable range hopping (VRH) consistently provides an Instead, a disordered TLL provides a natural explanation for accurate description of our data. VRH describes charge this unusual crossover from metallic to insulating behaviour. At transport by strongly-localized electrons: in a d-dimensional temperatures T\t , power-law behaviour in r(T) is a signature material rðTÞ¼ r exp½ðT =TÞ , where T is the characteristic > 0 0 of TLL behaviour in a q1D metal. Fitting rpT in the high- VRH temperature (which rises as the disorder increases) and temperature metallic regime of our crystals consistently yields n ¼ (1 þ d) . Although Mott’s original model assumed that 1oao1.01 (Fig. 2a). In a clean half-filled TLL, this would hopping occurred via inelastic electron–phonon scattering, VRH correspond to a Luttinger parameter K ¼ (a þ 3)/4B1, that is, has also been predicted to occur via e –e interactions in non-interacting electrons. However, disorder renormalizes disordered TLLs . the e –e interactions: for a commensurate chain of spinless Figure 3a displays VRH fits for crystals A–F, while fits to r(T) fermions, a ¼ 2K  2 and a critical point separates localized from in three further crystals which cracked during subsequent delocalized ground states at K ¼ 3/2 (ref. 6). Our experimental measurements are shown in Supplementary Fig. 5. All our values for a therefore indicate that Na Mo Se lies close to this crystals yield values for d ranging from 1.2 to 1.7 (Supplementary 2  d 6 6 critical point. Although the effects of incommensurate band Table I), in good agreement with the d ¼ 1.5 predicted for filling on a disordered TLL remain unclear, comparison with arrays of disordered conducting chains . Coulomb repulsion in clean TLLs suggests that removing electrons reduces K . For disordered materials opens a soft (quadratic) gap at E , leading to r F 1oK o3/2, r(T) is predicted to be metallic at high temperature, VRH transport with d ¼ 1 regardless of the actual dimensionality. before passing through a minimum at T (which rises with We consistently observe d41, implying that localized states are min NATURE COMMUNICATIONS | 7:12262 | DOI: 10.1038/ncomms12262 | www.nature.com/naturecommunications 3 Energy (eV) –1 –1 –1 N(E) (states eV spin u.c. ) ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12262 5E–6 A B C 4E–6 = 1.005±0.002  = 1.007±0.003  = 1.005±0.003 3E–6 2E–6 1E–6 0 100 200 0 100 200 0 100 200 300 T (K) T (K) T (K) 5E–6 D E F 4E–6 3E–6 = 1.008±0.005 = 1.004±0.002 2E–6 = 1.004±0.005 1E–6 0 100 200 0 100 200 300 100 200 300 T (K) T (K) T (K) b c 0.01 100 x 1E–4 min 1E–6 0 100 200 300 1E–7 1E–6 1E–5 T (K)  (300K) (Ωm) Figure 2 | Power laws and minima in the normal-state resistivity q(T). (a) r(T) for crystals A–F, together with power-law fits rpT (black lines, fitting range 1.5T oTo300 K). T corresponds to the minimum in r(T) for T4T .(b) r(T) plotted on a semi-logarithmic scale for crystals A and F: min min pk r E10 r as T-T .(c) Evolution of T with r(300 K), which is a measure of the disorder in each crystal. The horizontal shading indicates the F A pk min 1=ð1  zÞ estimated single-particle dimensional crossover temperature T B104 K, obtained using T  Wðt =WÞ , where W is the conduction bandwidth x x ? (Supplementary Note I), z ¼ðK þ K  2Þ=8 and K ¼ 3/2. No anomaly is visible in r(T)at T , suggesting either that T may be further renormalized due r r x x to competing charge instabilities , or that signatures of Tomonaga–Luttinger liquid behaviour may persist even for ToT (ref. 6). present at E and no gap develops in Na Mo Se . A small Further evidence for criticality is seen in the frequency F 2  d 6 6 paramagnetic contribution also emerges in the dc magnetization dependence of the conductivity s(o) within the divergent r(T) below T and rises non-linearly with 1/T (Supplementary regime (Fig. 3b). For crystals with sub-critical disorder, s(o) min Fig. 6). Similar behaviour has previously been attributed to a remains constant at low frequency, as expected for a disordered progressive crossover from Pauli to Curie paramagnetism due to metal. In contrast, s(o) in samples with super-critical disorder 2 2 electron localization (Supplementary Note IV). rises with frequency, following a o ln (1/o) trend. This is Although r(T) exhibits VRH divergence in all crystals prior to quantitatively compatible with both the Mott–Berezinskii formula peaking at T , a dramatic increase in r(T ) by 4 orders of for localized non-interacting electrons in 1D and the expected pk pk 6,31 magnitude occurs between crystals C and D. This is reminiscent behaviour of a disordered chain of interacting fermions . The of the rapid rise in resistivity on crossing the mobility edge in strong variation of s(o) even at sub-kHz frequencies implies that disordered 3D materials. Our data are therefore suggestive of a the localization length x is macroscopic, in contrast with the 1=d  1 crossover to strong localization and the existence of a critical x ðT N Þ t100 nm expected from Mott VRH theory . L E 0 F disorder or ‘q1D mobility edge’. Such behaviour may also However, it has been predicted that the relevant localization originate from proximity to the K ¼ 3/2 critical point. lengthscale for a weakly-disordered q1D crystal is the Larkin Interestingly, the critical disorder approximately correlates with (phase distortion) length, which may be exponentially large . the experimental condition T ET , where T is the estimated The evolution of the magnetoresistance (MR) r(H) with min x x single-particle dimensional crossover temperature (Fig. 2c). This temperature also supports a localization scenario. Above T , min suggests a possible role for dimensional crossover in establishing r(H) is weakly positive and follows the expected H dependence the mobility edge. for an open Fermi surface (Fig. 3c). At lower temperature, the 4 NATURE COMMUNICATIONS | 7:12262 | DOI: 10.1038/ncomms12262 | www.nature.com/naturecommunications (Ωm)  (Ωm)  (Ωm) T (K) min NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12262 ARTICLE 0.6 1E–5 A B C D E F 2.0E–5 2.0 6E–5 1.5 0.01 0.1 0.1 1.2E–6 1.5E–5 8E–6 3E–6 1.5E–5 0.01 9E–7 1.5 0.4 1E–3 1E–5 2E–6 1.0 6E–6 4E–5 0.01 1E–3 6E–7 1.0E–5 1.0 –ν 1E–4 –ν –ν –ν –ν –ν 4E–6 T T T 1E–4 T T 0.2 1E–6 5E–6 0.5 1E–3 2E–5 5.0E–6 0.5 2E–6 0.0 0.0 0.0 0 0.0 0 10203040 0 10203040 0 10203040 0 10203040 010 20 30 40 0 102030 40 T (K) T (K) T (K) T (K) T (K) T (K) b c d e T = 150K T = 10K T = 1.8K A (÷10) 1E5 100.0 100 100 100 8E4 D B C 6E4 80 99.9 R ∝ H 99.8 99.7 80 D 2 60 99.6 0 0 04 8 12 04 8 12 048 12 0 200 400 600 H (T) H (T) H (T) ⊥ ⊥ ⊥ Frequency (Hz) Figure 3 | Influence of electron localization on the low-temperature electrical transport. (a) Low-temperature divergence in the electrical resistivity r(T) for six Na Mo Se crystals A–F. Black lines are least-squares fits using a variable range hopping (VRH) model (Supplementary Note III). T (and hence 2  d 6 6 0 the disorder) rises monotonically from crystal A-F. Insets: r(T ) plotted on a semi-logarithmic scale; straight lines indicate VRH behaviour. (b) Frequency-dependent conductivity s(o) in crystals A, B, D and F (data points). Error bars correspond to the s.d. in the measured conductivity, that is, ðd þ 2Þ 2 30 our experimental noise level. For the highly-disordered crystals D and F, the black lines illustrate the o log ð1=oÞ trend predicted for strongly- localized electrons (using d ¼ 1). Data are acquired above T ,at T ¼ 4.9, 4.9, 4.6, 6 K for crystals A, B, D and F, respectively. (c–e) Normalized perpendicular pk magnetoresistance (MR) in crystal D (see Methods for details of the magnetic field orientation). At 150 K (c), the effects of disorder are weak and rpH due to the open Fermi surface. In the VRH regime at 10 K (d), magnetic fields delocalize electrons due to a Zeeman-induced change in the level occupancy , leading to a large negative MR. For ToT (e), the high-field MR is positive as superconductivity is gradually suppressed. The weak negative pk MR below H ¼ 3 T may be a signature of enhanced quasiparticle tunnelling: in a spatially-inhomogeneous superconductor, magnetic field-induced pair-breaking in regions where the superconducting order parameter is weak can increase the quasiparticle density and hence reduce the electrical resistance. MR data from crystal C are shown for comparison: here the disorder is lower and HB4 T destroys superconductivity. divergence in r(T) correlates with a crossover to strongly negative temperature (Supplementary Fig. 7, Supplementary Note VI). MR within the VRH regime (Fig. 3d). The presence of a soft We estimate an anisotropy x =x ¼ 6:0 in the coherence length, == ? Coulomb gap at E would lead to a positive MR within the VRH which is lower than the experimental values for Tl Mo Se and F 2 6 6 33 16 regime ; in contrast, our observed negative MR in Na Mo Se In Mo Se (13 and 17, respectively ) in spite of the smaller t in 2  d 6 6 2 6 6 > corresponds to a delocalization of gapless electronic states and Na Mo Se (Supplementary Fig. 2; see Methods for magnetic 2  d 6 6 provides additional evidence for a screened Coulomb interaction. field orientation details). This anisotropy is also far smaller than pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The MR switches sign again below T and becomes positive the measured conductivity ratio at 300 K: s =s ¼ 57. In pk == ? (Fig. 3e): as we shall now demonstrate, this is a signature of comparison, close agreement is obtained between the anisotropies superconductivity. in x and s for Li Mo O (ref. 39), where the effects of ==;? ==;? 0.9 6 17 disorder are believed to be weak . The disparate anisotropies in Na Mo Se arise from a strong suppression of x , thus 2  d 6 6 == illustrating the essential role of disorder in controlling the low- Superconducting transitions in Na Mo Se . The presence 2  d 6 6 15,16,35 temperature properties of Na Mo Se . of a superconducting ground state in Tl Mo Se and 2  d 6 6 2 6 6 In Mo Se implies that the peak in r(T) o6 K is likely to signify Although superconducting fluctuations are observed regardless 2 6 6 of the level of disorder in Na Mo Se , it is important to the onset of superconductivity in Na Mo Se . On cooling 2  d 6 6 2  d 6 6 crystals A–C in a dilution refrigerator, we uncover a identify whether phase-coherent long-range order develops in crystals D–F which exhibit super-critical disorder. In Fig. 4g–i, two-step superconducting transition characteristic of strongly 35–38 we demonstrate that r(T) in these samples still follows a 1D phase anisotropic q1D superconductors (Fig. 4a–c). Below T , pk slip model, albeit with a strongly enhanced contribution from superconducting fluctuations initially develop along individual quantum phase slips due to the increased disorder (Mo Se ) chains and r(T) is well-described by a 1D phase slip 6 6 N (Supplementary Note V). The fitting parameters for our 1D model (Supplementary Note V). Subsequently, a weak hump in phase slip analysis are listed in Supplementary Table II. A weak r(T) emerges (Fig. 4d–f) at temperatures ranging from B 0.95 K Meissner effect also develops in the magnetization below B 3.5 K (crystal A)toB 1.7 K (crystal C). This hump signifies the onset of in crystals D and E (Fig. 4g,h,j), but is rapidly suppressed by a transverse phase coherence due to inter-chain coupling. Cooper magnetic field. Low transverse phase stiffness is common in q1D pairs can now tunnel between the chains and a Meissner effect is superconductors: for example, bulk phase coherence in carbon expected to develop, but we are unable to observe this since 1.7 K lies below the operational range of our magnetometer. Analysis of nanotube arrays is quenched by 2–3 T, yet pairing persists up to 28 T . The superconducting volume fraction corresponding to the current–voltage characteristics indicates that a phase-coherent superconducting ground state is indeed established at low the magnitude of this Meissner effect is also unusually low: NATURE COMMUNICATIONS | 7:12262 | DOI: 10.1038/ncomms12262 | www.nature.com/naturecommunications 5 0.32 0.36 0.40 0.32 0.36 0.40 0.32 0.36 0.40 0.24 0.28 0.32 0.28 0.32 0.36 0.28 0.32 0.36 (Ωm) –1 (Sm ) –1 (Sm ) R(H) / R (%) max R(H) / R (%) max R(H) / R (%) max ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12262 a b c d e f 1E–4 1E–5 2E–5 6E–5 1E–5 8E–6 1E–5 4E–5 1E–5 6E–6 1E–6 1E–5 4E–6 2E–5 1E–6 1E–6 2E–6 1E–7 A B C A B C 0 0 0123 0123 0246 0.5 1.0 1.5 1.0 1.5 2.0 1.0 1.5 2.0 T (K) T (K) T (K) T (K) T (K) T (K) g j h i FC 0.1T 0.1 FC 0.1T 0 0 FC 0.1T 0.6 ZFC 2.0 ZFC 1.5 0.0 ZFC 0.5 –2 –2 FC 1.5 0.0025T –0.1 1.0 0.4 –4 –4 ZFC 1.0 –0.2 0.3 –6 –6 0.5 D E F E –0.3 0.5 2.0 2.5 0246 0246 0246 T (K) T (K) T (K) T (K) Figure 4 | Resistive and magnetic superconducting transitions in Na Mo Se . (a–c) Electrical resistivity r(To6 K) for crystals A–C. Coloured points 2  d 6 6 represent experimental data; black lines are fits to a 1D model incorporating thermal and quantum phase slips (Supplementary Note V). (d–f) Zoom views of r(T) in crystals A–C, plotted on a semi-logarithmic scale. The low-temperature limit of our 1D phase slip fits is signalled by a hump in r(T), highlighted by the transition from solid to dashed black fit lines: this corresponds to the onset of transverse phase coherence. In quasi-one-dimensional (q1D) superconductors, such humps form due to finite-size or current effects during dimensional crossover .(g–i) r(To6K) for the highly-disordered crystals D–F. Coloured points represent experimental data; black lines are fits to the same 1D phase slip model as in a–c, which accurately reproduces the broad superconducting transitions due to an increased quantum phase slip contribution (Supplementary Note V). Inhomogeneity and spatial fluctuations of the order parameter are expected to blur the characteristic hump in r(T) at dimensional crossover, thus explaining its absence from our data as the disorder rises. In g and h, we also plot zero-field-cooled/field-cooled (ZFC/FC) thermal hysteresis loops displaying the Meissner effect in the magnetic susceptibility w(T); j shows a zoom view of the susceptibility in crystal E. Data were acquired with the magnetic field parallel to the crystal c-axis and a paramagnetic background has been subtracted. The small diamagnetic susceptibilitiesjj w  1 are due to emergent pairing inhomogeneity creating isolated superconducting islands ; jj w is further decreased by the large magnetic penetration depth perpendicular to the c-axis in q1D crystals. o0.1%. Magnetic measurements of the superconducting volume Let us now examine the effects of disorder on the super- fraction in q1D materials invariably yield values o100%, since conducting ground state. Figure 5a illustrates T rising pk the magnetic penetration depth l normal to the 1D axis can monotonically from crystal A to F. Plotting T as a function of ab pk reach several microns and diamagnetic flux exclusion is r(300 K) (which is an approximate measure of the static disorder incomplete. For a typical Na Mo Se crystal of diameter in each crystal), we observe a step-like feature between crystals C 2  d 6 6 dB100mm, we estimate that a 0.1% volume fraction would and D, that is, at the critical disorder (Fig. 5b). Strikingly, the require l B10mm, which seems excessively large. Conversely, an characteristic VRH temperature T which we extract from our c 0 array of phase-fluctuating 1D superconducting filaments would r(T) fits (Fig. 3a) displays an identical dependence on r(300 K). not generate any Meissner effect at all. We therefore attribute the This implies that disorder controls both the superconducting unusually small Meissner signal to inhomogeneity in the ground state and the insulating tendency in r(T) at low superconducting order parameter, which is predicted temperature. The positive correlation between T and T pk 0 11,12,41,42 to emerge in the presence of intense disorder . (Fig. 5c) confirms that the onset temperature for superconducting In an inhomogeneous superconductor, Meissner screening is fluctuations (and hence the pairing energy D ) is enhanced by achieved via Josephson coupling between isolated super- localization in Na Mo Se . A concomitant increase in the 2  d 6 6 conducting islands . Within a single super-critically disordered transverse coherence temperature (Supplementary Note VI) Na Mo Se crystal, we therefore anticipate the formation implies that some enhancement in the phase stiffness also occurs. 2  d 6 6 of multiple Josephson-coupled networks comprising individual Super-critical disorder furthermore enables superconducting superconducting filaments. The total magnitude of the fluctuations to survive in high magnetic fields (Fig. 5d–g). In diamagnetic screening currents flowing percolatively through crystal C (which lies below the q1D mobility edge), super- each network will be much smaller than that in a homogeneous conductivity is completely quenched at all temperatures (that is, sample due to the smaller d/l ratio, thus diminishing the T -0) by H ¼ 4 T (Fig. 5d,f). A giant negative MR reappears for ab pk Meissner effect. H44 T (Fig. 3e), confirming that superconductivity originates from pairing between localized electrons. In contrast, the peak at r(T ) in the highly-disordered crystal F is strikingly resistant to pk magnetic fields (Fig. 5e,g): at T ¼ 4.6 K, our observed H ¼ 14 T, Enhancement of superconductivity by disorder. We have c2 established a clear influence of disorder on electrical transport in which exceeds the weak-coupling Pauli pair-breaking limit H ¼ 3 T by a factor 44 (see Supplementary Note VII for a Na Mo Se (Figs 2 and 3) and demonstrated that the peak in 2  d 6 6 P r(T)at T corresponds to the onset of superconductivity (Fig. 4). derivation of H (T)). A similar resilience is evident from the pk 6 NATURE COMMUNICATIONS | 7:12262 | DOI: 10.1038/ncomms12262 | www.nature.com/naturecommunications (Ωm) –3 Dimensionless susceptibility  ×10 (Ωm) (Ωm) –3 Dimensionless susceptibility  ×10 (Ωm) (Ωm) (Ωm) (Ωm) (Ωm) –3 Dimensionless susceptibility  ×10 (Ωm) NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12262 ARTICLE a b c 6 6 A B C D E F 1.0 1E4 0.9 1E3 0.8 1E2 23456 1E–7 1E–6 1E–5 1E2 1E3 1E4 T (K)  (300K) (Ωm) T (K) d e f C F C F 2.5 1.2E–4 4 T 14 T 2.0 1E–4 2 T 12 T 1 T 8 T 8E–5 0 T 1.5 0 T 6E–5 1.0 4E–5 0.5 2E–5 0 0.0 0 5 10 15 0 5 10 15 123 4.5 5.0 5.5 T (K) T (K) T (K) T (K) Figure 5 | Disorder controls the divergent electrical resistivity and enhances superconductivity. (a) Zoom view of the temperature-dependent electrical resistivity r(T) at the onset of superconductivity in all crystals, normalized to r(T ). (b) Evolution of the characteristic variable range hopping temperature pk T and the superconducting onset temperature T with r(300 K). The step at 10 O m corresponds to the critical disorder, that is, the quasi-one- 0 pk dimensional mobility edge. Error bars in r(300 K) are determined from the experimental noise level and our measurement resolution for the crystal dimensions. The error in T corresponds to its s.d., obtained from our variable range hopping fitting routine. (c) T versus T for each crystal, confirming 0 pk 0 the positive correlation between superconductivity and disorder. Data from three additional crystals which broke early during our series of measurements (Supplementary Note III) are also included (black circles). (d,e) Suppression of superconductivity with magnetic field H perpendicular to the c-axis for crystals C (d) and F (e). (f,g) Upper critical field H (T), equivalent to T (H), for crystals C (f) and F (g). Error bars in H (T) correspond to the error in c2 pk c2 determining the maximum in r(T,H). @H ðTÞ=@Tj ¼ 5:1TK and 24 T K for C and F, respectively. c2 pk positive MR in crystal D, which persists up to at least 14 T at 1.8 K that the localization length x remains larger than the coherence (Fig. 3e). Triplet pairing is unlikely to occur in Na Mo Se length (that is, the Cooper pair radius). However, experiments 2  d 6 6 (since scattering would rapidly suppress a nodal order parameter) have invariably shown superconductivity to be destroyed and orbital limiting is also suppressed (since vortices cannot form by disorder, due to enhanced Coulomb pair-breaking , phase 42,46,47 10,48 across phase-incoherent filaments). Our data therefore suggest fluctuations or emergent spatial inhomogeneity . that disorder lifts H , creating anomalously strong correlations In particular, increasing disorder in Li Mo O (one of the P 0.9 6 17 which raise the pairing energy D (refs 10,11) above the weak- few q1D superconductors extensively studied in the literature) coupling 1.76 k T . A direct spectroscopic technique would be monotonically suppresses superconductivity . Therefore, the key B pk required to determine the absolute enhancement of D , since question arising from our work is why the onset temperature for spin-orbit scattering from the heavy Mo ions will also contribute superconductivity rises with disorder in Na Mo Se ,in 2  d 6 6 to raising H . contrast to all other known materials? Disorder acts to enhance the matrix element for e –e interactions. This may be explained qualitatively by considering Discussion that all conduction electron wavefunctions experience the same disorder-induced potential, developing inhomogeneous multi- The emergence of a superconducting ground state in Na Mo Se places further constraints on the origin of the fractal probability densities and hence becoming spatially 2  d 6 6 correlated. Such enhanced correlations have been predicted to normal-state divergence in r(T). Our electronic structure calculations indicate that the q1D Fermi surface of Na Mo Se increase the Cooper pairing energy : in the absence of pair- 2  d 6 6 breaking by long-ranged Coulomb interactions, this will lead to a is almost perfectly nested: any incipient electronic DW would 11–13,51,52 therefore gap the entire Fermi surface, creating clear signatures of rise in the superconducting transition temperature . A proposal to observe this effect in superconducting hetero- a gap in r(T) and leaving no electrons at E to form a superconducting condensate. In contrast, our VRH fits and MR structures with built-in Coulomb screening (by depositing superconducting thin films on substrates with high dielectric data do not support the formation of a DW gap, and a superconducting transition occurs at low temperature. Electrons constants) has not yet been experimentally realised. However, our VRH dimensionality d41 (Fig. 3a) and negative MR (Fig. 3d,e) must therefore remain at E for all T4T , indicating that r(T) F pk diverges due to disorder-induced localization rather than any both point towards a weak or screened Coulomb repulsion, while other insulating instability. the power laws and broad minima in r(T) at high temperature (Fig. 2a) indicate a Luttinger parameter K 41. These results all It has been known since the 1950s that an s-wave super- 44,45 conducting order parameter is resilient to disorder , provided imply that e –e interactions in Na Mo Se are attractive. 2  d 6 6 NATURE COMMUNICATIONS | 7:12262 | DOI: 10.1038/ncomms12262 | www.nature.com/naturecommunications 7 /(T ) (Ωm) pk (Ωm) T (K) H (T) c2 T (K) pk T (K) pk H (T) c2 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12262 (For comparison, K B0.25 in Li Mo O and the e –e Beyond enhancing superconductivity, the ability to simulta- r 0.9 6 17 7,8 interaction is repulsive .) Phonon-mediated coupling—the neously modulate band filling, disorder and dimensionality Cooper channel—therefore appears to dominate over the promises a high level of control over emergent order, including Coulomb repulsion in Na Mo Se , suggesting that the usual DWs and magnetic phases. More generally, Na Mo Se 2  d 6 6 2  d 6 6 disorder-induced Coulomb pair-breaking may be avoided. Below and other similar interrupted strand materials may be ideal the q1D mobility edge, our rise in T is quantitatively compatible environments in which to study the evolution of many-body pk with a weak multifractal scenario (Supplementary Fig. 8, electron localization beyond the non-interacting Anderson limit. Supplementary Note VIII), providing a possible explanation for the enhancement of superconductivity which merits further Methods theoretical attention. Crystal growth and initial characterization. A series of Na Mo Se crystals 2  d 6 6 The fact that no experimental examples of q1D materials with was grown using a solid-state synthesis procedure. The precursor materials were MoSe , InSe, Mo and NaCl, all in powder form. Before use, the Mo powder was attractive e –e interactions have yet been reported poses the 2 reduced under H gas flowing at 1,000 C for 10 h, to eliminate any trace of oxygen. question why Na Mo Se should be different. Although strong 2  d 6 6 The MoSe was prepared by reacting Se with H -reduced Mo in a ratio 2:1 inside a 2 2 electron–phonon coupling is known to play an important role in purged, evacuated and flame-baked silica tube (with a residual pressure of 16,53 the physics of molybdenum cluster compounds , we propose B10 mbar argon), which was then heated to B700 C for 2 days. InSe was synthesized from elemental In and Se in an evacuated sealed silica tube at 800 C that the disordered q1D nature of Na Mo Se may instead play 2  d 6 6 for 1 day. Powder samples of Na Mo Se were prepared in two steps. First, 2  d 6 6 the dominant role, by suppressing the Coulomb repulsion. In the In Mo Se was synthesized from a stoichiometric mixture of InSe, MoSe and Mo, 2 6 6 2 presence of disorder, a q1D material can be regarded as a parallel heated to 1,000C in an evacuated sealed silica tube for 36 h. Second, an ion array of ‘interrupted strands’ , that is, a bundle of finite-length exchange reaction of In Mo Se with NaCl was performed at 800 C, using a 10% 2 6 6 nanowires. The electric polarizability of metallic nanoparticles is NaCl excess to ensure total exchange as described in ref. 61. All starting reagents were found to be monophase on the basis of their powder XRD patterns, acquired strongly enhanced relative to bulk materials , although this effect using a D8 Bruker Advance diffractometer equipped with a LynxEye detector is usually cancelled out by self-depolarization. The geometric (CuKa radiation). Furthermore, to avoid any contamination by oxygen and depolarization factor vanishes for q1D symmetry, leading to giant moisture, the starting reagents were kept and handled in a purified argon-filled dielectric constants e which rise as the filament length increases . glovebox. To synthesize single crystals, a Na Mo Se powder sample (of mass B5g) This effect was recently observed in Au nanowires , with e 2  d 6 6 7 was cold-pressed and loaded into a molybdenum crucible, which had previously reaching 10 .InNa Mo Se , we therefore anticipate that 2  d 6 6  5 been outgassed at 1,500 C for 15 min under a dynamic vacuum of B10 mbar. the long-range Coulomb repulsion in an individual (Mo Se ) 6 6 l The Mo crucible was subsequently sealed under a low argon pressure using an filament (loN) will be efficiently screened by neighbouring arc-welding system. The Na Mo Se powder charge was heated at a rate of 2  d 6 6 300 Ch up to 1,750 C, held at this temperature for 3 h, then cooled at filaments . This intrinsic screening provides a natural 100 Ch down to 1,000 C and finally cooled naturally to room temperature explanation for attractive e –e interactions and suppresses within the furnace. Crystals obtained using this procedure have a needle-like shape Coulomb pair-breaking in the superconducting phase. with length up to 4 mm and a hexagonal cross-section with typical diameter It has been suggested that impurities can increase the r150mm. Initial semi-quantitative microanalyses using a JEOL JSM 6400 scanning electron microscope equipped with an Oxford INCA EDX spectrometer indicated temperature at which transverse phase coherence is established that the Na contents ranged between 1.7 and 2, that is, up to 15% deficiency. The in q1D superconductors . This effect cannot be responsible for Na deficiency results from the high temperatures used during the crystal growth our observed rise in T , which corresponds to the onset of 1D pk process coupled with the small size of the Na ion: it cannot be accurately controlled superconducting fluctuations on individual (Mo Se ) filaments. 6 6 l within the conditions necessary for crystal growth. Since In Mo Se is known to be superconducting below 2.85 K , it is important We also point out that the finite-size effects which influence 2 6 6 59 60 to consider the possibility of In contamination in our samples. The Na/In ion critical temperatures in granular or nanomaterials are not 61,62 exchange technique used during synthesis is known to be highly efficient and relevant in Na Mo Se : quantum confinement is absent in 2  d 6 6 In Mo Se decomposes above 1,300 C, well below our crystal growth temperature 2 6 6 homogeneously-disordered crystalline superconductors and (1,750 C). This precludes the presence of any superconducting In Mo Se 2 6 6 hence no peaks form in N(E ). These mechanisms are discussed (or In-rich (In,Na) Mo Se ) filaments in our crystals. Diffuse X-ray scattering F 2 6 6 measurements accordingly reveal none of the Huang scattering or disk-like Bragg in detail in Supplementary Note IX. reflections which would be produced by such filaments. Furthermore, EDX In summary, we have presented experimental evidence for the spectrometry is unable to detect any In content in our crystals, while inductively- enhancement of superconductivity by disorder in Na Mo Se . 2  d 6 6 coupled plasma mass spectrometry indicates a typical In residual of o0.01%, that The combination of q1D crystal symmetry (and the associated is, o0.0002 In atoms per unit cell. The electronic properties of Na Mo Se 2  d 6 6 crystals will remain unaffected by such a tiny In residual in solid solution. dimensional crossover), disorder and incommensurate band filling in this material poses a challenge to existing 1D/q1D theoretical models. Although the normal-state electrical resistivity Electrical transport measurements. Before all measurements, the as-grown of Na Mo Se is compatible with theories for disordered 1D crystal surfaces were briefly cleaned with dilute hydrochloric acid (to remove any 2  d 6 6 residue from the Mo crucible and hence minimize the contact resistance), followed systems with attractive e –e interactions, we establish several by distilled water, acetone and ethanol. Four Au contact pads were sputtered onto unusual low-temperature transport properties which deserve the upper surface and sides of each crystal using an Al foil mask; 50mm Au wires future attention. These include a resistivity which diverges were then glued to these pads using silver-loaded epoxy cured at 70 C (Epotek following a q1D VRH law for all levels of disorder, the existence E4110). Special care was taken to thoroughly coat each end of the crystal with epoxy, to ensure that the measurement current passed through the entire crystal. of a critical disorder or q1D mobility edge where T ET , and a min x 2 All contacts were verified to be Ohmic at room temperature before and after each strongly frequency-dependent conductivity s(o)Bo in crystals series of transport measurements, and at T ¼ 4 K after cooling. Typical contact with super-critical disorder. At temperature T , 1D super- pk resistances were of the order of 2O at 300 K. The transverse conductivity s was conducting fluctuations develop, and a phase-coherent ground estimated at room temperature using a four-probe technique, with contacts on opposite hexagonal faces of a single crystal. The temperature dependence of the state is established via coupling between 1D filaments at lower transverse resistivity r (T) has never been accurately measured in M Mo Se due > 2 6 6 temperature. As the disorder rises, T increases: in our most- pk to the exceptionally large anisotropies, small crystal diameters and high fragility, disordered crystals, the survival of superconducting fluctuations 15 even in the least anisotropic Tl Mo Se which forms the largest crystals . 2 6 6 in magnetic fields at least four times larger than the Pauli limit Low-frequency four-wire ac conductivity measurements were performed in two separate cryogen-free systems: a variable temperature cryostat and a dilution suggests that the pairing energy may be unusually large. refrigerator, both of which may be used in conjunction with a superconducting We conclude that deliberately introducing disorder into q1D vector magnet. The ac conductivity was measured using a Keithley 6100 current crystals represents a new path towards engineering correlated source, a Stanford SRS850 lock-in amplifier with input impedance 10 MO and (for electron materials, in remarkable contrast with the conventional low resistances, that is, weakly-disordered samples) a Stanford SR550 preamplifier blend of strong Coulomb repulsion and a high density of states. with input impedance 100 MO. Data from several crystals were cross-checked using 8 NATURE COMMUNICATIONS | 7:12262 | DOI: 10.1038/ncomms12262 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12262 ARTICLE a Quantum Design Physical Property Measurement System with the standard 19. Narduzzo, A., Enayati-Rad, A., Horii, S. & Hussey, N. E. Possible coexistence of inbuilt ac transport hardware: both methods generate identical, reproducible data. local itinerancy and global localization in a quasi-one-dimensional conductor. 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Superconductivity at 6K and the violation of Pauli limit in 27.6 kO, respectively: the crystal diameter increases from D to F, thus explaining Ta Pd S . J. Phys. Soc. Jpn. 83, 023702 (2014). the rise in resistivity despite a fall in resistance. These values remain much smaller 2 x 5 23. dos Santos, C. et al. Electrical transport in single-crystalline Li Mo O : 0.9 6 17 than our lock-in amplifier input impedance, ruling out any current leakage in a two-band Luttinger liquid exhibiting Bose metal behavior. Phys. Rev. B 77, highly-disordered crystals. Our measurement current I ¼ 10mA leads to a ac 193106 (2008). maximum power dissipation o10mW. This is negligible compared with the B2 mW cooling power at 2 K on our cryostat cold finger and we may hence rule 24. dos Santos, C., White, B., Yu, Y.-K., Neumeier, J. & Souza, J. Dimensional out any sample heating effects in our data. crossover in the purple bronze Li Mo O . 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Acknowledgements To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ We thank Alexei Bosak (Beamline ID28, ESRF Grenoble) for assistance with data collection and processing, and Igor Burmistrov, Vladimir Kravtsov, Tomi Ohtsuki and Vincent Sacksteder IV for stimulating discussions. The Swiss-Norwegian Beamlines r The Author(s) 2016 10 NATURE COMMUNICATIONS | 7:12262 | DOI: 10.1038/ncomms12262 | www.nature.com/naturecommunications http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Nature Communications Springer Journals

A disorder-enhanced quasi-one-dimensional superconductor

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Springer Journals
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Copyright © 2016 by The Author(s)
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Science, Humanities and Social Sciences, multidisciplinary; Science, Humanities and Social Sciences, multidisciplinary; Science, multidisciplinary
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2041-1723
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10.1038/ncomms12262
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Abstract

ARTICLE Received 26 Oct 2015 | Accepted 17 Jun 2016 | Published 22 Jul 2016 DOI: 10.1038/ncomms12262 OPEN A disorder-enhanced quasi-one-dimensional superconductor 1, 1, 2 3 4,5 4 4 A.P. Petrovic´ *, D. Ansermet *, D. Chernyshov , M. Hoesch , D. Salloum , P. Gougeon , M. Potel , 6 1 L. Boeri & C. Panagopoulos A powerful approach to analysing quantum systems with dimensionality d41 involves adding a weak coupling to an array of one-dimensional (1D) chains. The resultant quasi-1D (q1D) systems can exhibit long-range order at low temperature, but are heavily influenced by interactions and disorder due to their large anisotropies. Real q1D materials are therefore ideal candidates not only to provoke, test and refine theories of strongly correlated matter, but also to search for unusual emergent electronic phases. Here we report the unprecedented enhancement of a superconducting instability by disorder in single crystals of Na Mo Se , 2  d 6 6 a q1D superconductor comprising MoSe chains weakly coupled by Na atoms. We argue that disorder-enhanced Coulomb pair-breaking (which usually destroys superconductivity) may be averted due to a screened long-range Coulomb repulsion intrinsic to disordered q1D materials. Our results illustrate the capability of disorder to tune and induce new correlated electron physics in low-dimensional materials. Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 637371 Singapore. 2 3 Swiss-Norwegian Beamlines, European Synchrotron Radiation Facility, 6 rue Jules Horowitz, F-38043 Grenoble Cedex, France. Diamond Light Source, Harwell Campus, Didcot OX11 0DE, Oxfordshire, UK. Sciences Chimiques, CSM UMR CNRS 6226, Universite´ de Rennes 1, Avenue du Ge´ne´ral Leclerc, 5 6 35042 Rennes Cedex, France. Faculty of Science III, Lebanese University, PO Box 826, Kobbeh-Tripoli, Lebanon. Institute for Theoretical and Computational Physics, TU Graz, Petersgasse 16, 8010 Graz, Austria. * These authors contributed equally to this work. Correspondence and requests for materials should be addressed to A.P.P. (email: appetrovic@ntu.edu.sg) or to C.P. (email: christos@ntu.edu.sg). NATURE COMMUNICATIONS | 7:12262 | DOI: 10.1038/ncomms12262 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12262 eakly-interacting electrons in a three-dimensional high polarizability of disordered q1D materials. The combination (3D) periodic potential are well-described by of disorder and q1D crystal symmetry constitutes a new recipe WLandau–Fermi liquid theory , in which the free for strongly correlated electron liquids with tunable electronic electrons of a Fermi gas become dressed quasiparticles with properties. renormalized dynamical properties. Conversely, in the one- dimensional (1D) limit a Tomonaga–Luttinger liquid (TLL) is Results 2,3 formed , where single-particle excitations are replaced by highly Crystal and electronic structure of Na Mo Se .Na Mo Se 2  d 6 6 2  d 6 6 correlated collective excitations. So far, it has proved difficult to belongs to the q1D M Mo Se family (M ¼ Group IA alkali 2 6 6 interpolate theoretically between these two regimes, either by metals, Tl, In) which crystallize with hexagonal space group strengthening electron–electron (e –e ) interactions in 3D, or P6 /m. The structure can be considered as a linear condensation 4,5 by incorporating weak transverse coupling into 1D models . The of Mo Se clusters into infinite-length (Mo Se ) chains parallel 6 8 6 6 N invariable presence of disorder in real materials places further to the hexagonal c-axis, weakly coupled by M atoms (Fig. 1a). The demands on theory, particularly in the description of ordered q1D nature of these materials is apparent from the needle-like electronic ground states. Q1D systems such as nanowire ropes, morphology of as-grown crystals (Fig. 1b; see Methods for growth filamentary networks or single crystals with uniaxial anisotropy details). Ab initio calculations (Supplementary Note I) using therefore represent an opportunity to experimentally probe what density functional theory reveal an electronic structure which is theories aspire to model: strongly correlated electrons subject to uniquely simple among q1D metals. A single spin-degenerate disorder in a highly anisotropic 3D environment. band of predominant Mo d character crosses the Fermi energy xz Physical properties of q1D materials may vary considerably E at half-filling (Fig. 1c, Supplementary Fig. 1), creating a 1D with temperature. TLL theory is expected to be valid at Fermi surface composed of two sheets lying close to the Brillouin elevated temperatures, since electrons cannot hop coherently zone boundaries at p/c (where c is the c-axis lattice parameter). perpendicular to the high-symmetry axis and q1D systems behave The warping of these sheets (and hence the coupling between as decoupled arrays of 1D filaments. Phase-coherent single- (Mo Se ) chains) is controlled by the M cation, yielding 6 6 N particle hopping can only occur below temperature T rt x > values for t ranging from 230 K (M ¼ Tl) to 30 K (M ¼ Rb) (where t is the transverse hopping integral), at which a (Supplementary Fig. 2). In addition to tuning the dimensionality, dimensional crossover to an anisotropic quasi-3D (q3D) electron M also controls the ground state: M ¼ Tl, In are 4,6 liquid is anticipated . The properties of such q3D liquids remain 15,16 superconductors , while M ¼ K, Rb become insulating at low largely unknown, especially the role of electronic correlations in 16,17 temperature . determining the ground state. At low temperature, a TLL is Within the M Mo Se family, M ¼ Na is attractive for two 2 6 6 unstable to either density wave (DW) or superconducting reasons. First, we calculate an intermediate t ¼ 120 K, suggesting fluctuations, depending on whether the e –e interaction that Na Mo Se lies at the threshold between superconducting 2  d 6 6 is repulsive (due to Coulomb forces) or attractive (from and insulating instabilities. Second, the combination of the small electron–phonon coupling). Following dimensional crossover, Na cation size and a high growth temperature (1750 C) results in the influence of such interactions in the q3D state is unclear. substantial Na vacancy formation during crystal synthesis. Since As an example, electrical transport in the TLL state of the Na atoms are a charge reservoir for the (Mo Se ) chains, 6 6 N the q1D purple bronze Li Mo O is dominated by repulsive 0.9 6 17 these vacancies will reduce E and lead to an incommensurate 7,8 e –e interactions , yet a superconducting transition occurs band filling. Despite the reduction in carrier density, the density for temperatures below 1.9 K. of states N(E ) remains constant for Na (Fig. 1d, F 1.5-2.1 Disorder adds further complication to q1D materials due Supplementary Note I). Energy-dispersive X-ray (EDX) spectro- to its tendency to localize electrons at low temperature. For metry on our crystals indicates Na contents from 1.7 to 2, dimensionality dr2, localization occurs for any non-zero comfortably within this range. This is confirmed by synchrotron disorder; in contrast, for d42 a critical disorder is required and X-ray diffraction (XRD) on three randomly-chosen crystals: a mobility edge separates extended from localized states. ± ± structural refinements reveal Na deficiencies of 11 1%, 11 2% The question of whether a mobility edge can form in q1D and 13 4% (that is, d ¼ 0.22, 0.22, 0.26), but the (Mo Se ) 6 6 N materials after crossover to a q3D liquid state is open, as is the chains remain highly ordered. No deviation from the M Mo Se 2 6 6 microscopic nature of the localized phase. Disorder also structure is observed between 293 and 20 K, ruling out any lattice renormalizes e –e interactions, leading to a dynamic ampli- distortions such as the Peierls transition, which often afflicts q1D fication of the Coulomb repulsion and a weaker enhancement of metals. To probe the Na vacancy distribution, we perform diffuse 10–13 phonon-mediated e –e attraction, that is, Cooper pairing . X-ray scattering experiments on the d ¼ 0.26 crystal. No trace of We therefore anticipate that disorder should strongly suppress any Huang scattering (from clustered Na vacancies) or structured superconductivity in q1D materials, unless the Coulomb diffuse scattering from short-range vacancy ordering is observed interaction is unusually weak or screened. (Supplementary Fig. 3, Supplementary Note II). Na vacancies In this work, we show that the q1D superconductor therefore create an intrinsic, random disorder potential in Na Mo Se provides a unique environment in which to study 2  d 6 6 Na Mo Se single crystals. 2  d 6 6 the interplay between dimensionality, electronic correlations and disorder. Although Na Mo Se is metallic at room tempera- 2  d 6 6 ture, the presence of Na vacancy disorder leads to electron Normal-state electrical transport. We first examine the electrical localization and a divergent resistivity r(T) at low temperature, transport at high energy for signatures of disorder and one- prior to a superconducting transition. In contrast with all other dimensionality. The temperature dependence of the resistivity known superconductors, the onset temperature for super- r(T) for six randomly-selected Na Mo Se crystals A–F is 2  d 6 6 conducting fluctuations T is positively correlated with the level shown in Fig. 2a. r(300 K) increases by 41 order of magnitude pk of disorder. Normal-state electrical transport measurements also from crystal A to F (Fig. 2b): such large differences between display signatures of an attractive e –e interaction, which is crystals cannot be attributed to changes in the carrier density due consistent with disorder-enhanced superconductivity. A plausible to Na stoichiometry variation and must instead arise from explanation for these phenomena is an intrinsic screening of the disorder. Despite the variance in r(300 K), the evolution of r(T) long-range Coulomb repulsion in Na Mo Se , arising from the is qualitatively similar in all crystals. On cooling, r(T) exhibits 2  d 6 6 2 NATURE COMMUNICATIONS | 7:12262 | DOI: 10.1038/ncomms12262 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12262 ARTICLE a b Na Mo Se c d 0.0 E –0.4 –0.8 –1.2 Γ MK Γ AL H A –2 –1 0 E–E (eV) Figure 1 | Quasi-one-dimensional crystal and electronic structures in Na Mo Se . (a) Hexagonal crystal structure of Na Mo Se , viewed 2  d 6 6 2  d 6 6 perpendicular and parallel to the c-axis. From synchrotron X-ray diffraction experiments, we measure the a- and c-axis lattice parameters to be 8.65 Å and 4.49 Å, respectively at 293 K (Supplementary Note II). (b) Electron micrograph of a typical Na Mo Se crystal. Scale bar, 300mm. (c) Calculated 2  d 6 6 energy-momentum dispersion of the conduction band within the hexagonal Brillouin zone, highlighting the large bandwidth and minimal dispersion perpendicular to the chain axis. (d) Electronic density of states N(E) around the Fermi level in Na Mo Se . 2 6 6 metallic behaviour before passing through a broad minimum at increasing disorder) and diverging at lower temperature. These T and diverging at lower temperature. T falls from 150 K to features are consistently reproduced in our data. min min B70 K as r(300 K) decreases (Fig. 2c), suggesting that the Within the disordered TLL paradigm, our high-temperature divergence in r(T) and the disorder level are linked. transport data indicate that the e –e interaction is Upturns or divergence in r(T) have been widely reported in attractive, that is, K 41. This implies that electron–phonon 18–22 q1D materials and variously attributed to localization , coupling dominates over Coulomb repulsion and suggests that 23 24,25 multiband TLL physics , DW formation , incipient density the Coulomb interaction may be intrinsically screened in 16 8 fluctuations and proximity to Mott instabilities . Differentiating Na Mo Se . A quantitative analysis of the low-temperature 2  d 6 6 between these mechanisms has proved challenging, in part due to divergence in r(T) provides further support for the influence of the microscopic similarity between localized electrons and disorder, as well as a weak/screened Coulomb repulsion. We have randomly-pinned DWs in 1D. We briefly remark that the attempted to fit r(T) using a wide variety of resistive mechanisms: broad minimum in r(T)inNa Mo Se contrasts strongly with gap formation (Arrhenius activation), repulsive TLL power 2  d 6 6 the abrupt jumps in r(T) for nesting-driven DW materials such laws, weak and strong localization (Supplementary Fig. 4, as NbSe (ref. 26), while any Mott transition will be suppressed Supplementary Note III). Among these models, only Mott due to the non-stoichiometric Na content. variable range hopping (VRH) consistently provides an Instead, a disordered TLL provides a natural explanation for accurate description of our data. VRH describes charge this unusual crossover from metallic to insulating behaviour. At transport by strongly-localized electrons: in a d-dimensional temperatures T\t , power-law behaviour in r(T) is a signature material rðTÞ¼ r exp½ðT =TÞ , where T is the characteristic > 0 0 of TLL behaviour in a q1D metal. Fitting rpT in the high- VRH temperature (which rises as the disorder increases) and temperature metallic regime of our crystals consistently yields n ¼ (1 þ d) . Although Mott’s original model assumed that 1oao1.01 (Fig. 2a). In a clean half-filled TLL, this would hopping occurred via inelastic electron–phonon scattering, VRH correspond to a Luttinger parameter K ¼ (a þ 3)/4B1, that is, has also been predicted to occur via e –e interactions in non-interacting electrons. However, disorder renormalizes disordered TLLs . the e –e interactions: for a commensurate chain of spinless Figure 3a displays VRH fits for crystals A–F, while fits to r(T) fermions, a ¼ 2K  2 and a critical point separates localized from in three further crystals which cracked during subsequent delocalized ground states at K ¼ 3/2 (ref. 6). Our experimental measurements are shown in Supplementary Fig. 5. All our values for a therefore indicate that Na Mo Se lies close to this crystals yield values for d ranging from 1.2 to 1.7 (Supplementary 2  d 6 6 critical point. Although the effects of incommensurate band Table I), in good agreement with the d ¼ 1.5 predicted for filling on a disordered TLL remain unclear, comparison with arrays of disordered conducting chains . Coulomb repulsion in clean TLLs suggests that removing electrons reduces K . For disordered materials opens a soft (quadratic) gap at E , leading to r F 1oK o3/2, r(T) is predicted to be metallic at high temperature, VRH transport with d ¼ 1 regardless of the actual dimensionality. before passing through a minimum at T (which rises with We consistently observe d41, implying that localized states are min NATURE COMMUNICATIONS | 7:12262 | DOI: 10.1038/ncomms12262 | www.nature.com/naturecommunications 3 Energy (eV) –1 –1 –1 N(E) (states eV spin u.c. ) ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12262 5E–6 A B C 4E–6 = 1.005±0.002  = 1.007±0.003  = 1.005±0.003 3E–6 2E–6 1E–6 0 100 200 0 100 200 0 100 200 300 T (K) T (K) T (K) 5E–6 D E F 4E–6 3E–6 = 1.008±0.005 = 1.004±0.002 2E–6 = 1.004±0.005 1E–6 0 100 200 0 100 200 300 100 200 300 T (K) T (K) T (K) b c 0.01 100 x 1E–4 min 1E–6 0 100 200 300 1E–7 1E–6 1E–5 T (K)  (300K) (Ωm) Figure 2 | Power laws and minima in the normal-state resistivity q(T). (a) r(T) for crystals A–F, together with power-law fits rpT (black lines, fitting range 1.5T oTo300 K). T corresponds to the minimum in r(T) for T4T .(b) r(T) plotted on a semi-logarithmic scale for crystals A and F: min min pk r E10 r as T-T .(c) Evolution of T with r(300 K), which is a measure of the disorder in each crystal. The horizontal shading indicates the F A pk min 1=ð1  zÞ estimated single-particle dimensional crossover temperature T B104 K, obtained using T  Wðt =WÞ , where W is the conduction bandwidth x x ? (Supplementary Note I), z ¼ðK þ K  2Þ=8 and K ¼ 3/2. No anomaly is visible in r(T)at T , suggesting either that T may be further renormalized due r r x x to competing charge instabilities , or that signatures of Tomonaga–Luttinger liquid behaviour may persist even for ToT (ref. 6). present at E and no gap develops in Na Mo Se . A small Further evidence for criticality is seen in the frequency F 2  d 6 6 paramagnetic contribution also emerges in the dc magnetization dependence of the conductivity s(o) within the divergent r(T) below T and rises non-linearly with 1/T (Supplementary regime (Fig. 3b). For crystals with sub-critical disorder, s(o) min Fig. 6). Similar behaviour has previously been attributed to a remains constant at low frequency, as expected for a disordered progressive crossover from Pauli to Curie paramagnetism due to metal. In contrast, s(o) in samples with super-critical disorder 2 2 electron localization (Supplementary Note IV). rises with frequency, following a o ln (1/o) trend. This is Although r(T) exhibits VRH divergence in all crystals prior to quantitatively compatible with both the Mott–Berezinskii formula peaking at T , a dramatic increase in r(T ) by 4 orders of for localized non-interacting electrons in 1D and the expected pk pk 6,31 magnitude occurs between crystals C and D. This is reminiscent behaviour of a disordered chain of interacting fermions . The of the rapid rise in resistivity on crossing the mobility edge in strong variation of s(o) even at sub-kHz frequencies implies that disordered 3D materials. Our data are therefore suggestive of a the localization length x is macroscopic, in contrast with the 1=d  1 crossover to strong localization and the existence of a critical x ðT N Þ t100 nm expected from Mott VRH theory . L E 0 F disorder or ‘q1D mobility edge’. Such behaviour may also However, it has been predicted that the relevant localization originate from proximity to the K ¼ 3/2 critical point. lengthscale for a weakly-disordered q1D crystal is the Larkin Interestingly, the critical disorder approximately correlates with (phase distortion) length, which may be exponentially large . the experimental condition T ET , where T is the estimated The evolution of the magnetoresistance (MR) r(H) with min x x single-particle dimensional crossover temperature (Fig. 2c). This temperature also supports a localization scenario. Above T , min suggests a possible role for dimensional crossover in establishing r(H) is weakly positive and follows the expected H dependence the mobility edge. for an open Fermi surface (Fig. 3c). At lower temperature, the 4 NATURE COMMUNICATIONS | 7:12262 | DOI: 10.1038/ncomms12262 | www.nature.com/naturecommunications (Ωm)  (Ωm)  (Ωm) T (K) min NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12262 ARTICLE 0.6 1E–5 A B C D E F 2.0E–5 2.0 6E–5 1.5 0.01 0.1 0.1 1.2E–6 1.5E–5 8E–6 3E–6 1.5E–5 0.01 9E–7 1.5 0.4 1E–3 1E–5 2E–6 1.0 6E–6 4E–5 0.01 1E–3 6E–7 1.0E–5 1.0 –ν 1E–4 –ν –ν –ν –ν –ν 4E–6 T T T 1E–4 T T 0.2 1E–6 5E–6 0.5 1E–3 2E–5 5.0E–6 0.5 2E–6 0.0 0.0 0.0 0 0.0 0 10203040 0 10203040 0 10203040 0 10203040 010 20 30 40 0 102030 40 T (K) T (K) T (K) T (K) T (K) T (K) b c d e T = 150K T = 10K T = 1.8K A (÷10) 1E5 100.0 100 100 100 8E4 D B C 6E4 80 99.9 R ∝ H 99.8 99.7 80 D 2 60 99.6 0 0 04 8 12 04 8 12 048 12 0 200 400 600 H (T) H (T) H (T) ⊥ ⊥ ⊥ Frequency (Hz) Figure 3 | Influence of electron localization on the low-temperature electrical transport. (a) Low-temperature divergence in the electrical resistivity r(T) for six Na Mo Se crystals A–F. Black lines are least-squares fits using a variable range hopping (VRH) model (Supplementary Note III). T (and hence 2  d 6 6 0 the disorder) rises monotonically from crystal A-F. Insets: r(T ) plotted on a semi-logarithmic scale; straight lines indicate VRH behaviour. (b) Frequency-dependent conductivity s(o) in crystals A, B, D and F (data points). Error bars correspond to the s.d. in the measured conductivity, that is, ðd þ 2Þ 2 30 our experimental noise level. For the highly-disordered crystals D and F, the black lines illustrate the o log ð1=oÞ trend predicted for strongly- localized electrons (using d ¼ 1). Data are acquired above T ,at T ¼ 4.9, 4.9, 4.6, 6 K for crystals A, B, D and F, respectively. (c–e) Normalized perpendicular pk magnetoresistance (MR) in crystal D (see Methods for details of the magnetic field orientation). At 150 K (c), the effects of disorder are weak and rpH due to the open Fermi surface. In the VRH regime at 10 K (d), magnetic fields delocalize electrons due to a Zeeman-induced change in the level occupancy , leading to a large negative MR. For ToT (e), the high-field MR is positive as superconductivity is gradually suppressed. The weak negative pk MR below H ¼ 3 T may be a signature of enhanced quasiparticle tunnelling: in a spatially-inhomogeneous superconductor, magnetic field-induced pair-breaking in regions where the superconducting order parameter is weak can increase the quasiparticle density and hence reduce the electrical resistance. MR data from crystal C are shown for comparison: here the disorder is lower and HB4 T destroys superconductivity. divergence in r(T) correlates with a crossover to strongly negative temperature (Supplementary Fig. 7, Supplementary Note VI). MR within the VRH regime (Fig. 3d). The presence of a soft We estimate an anisotropy x =x ¼ 6:0 in the coherence length, == ? Coulomb gap at E would lead to a positive MR within the VRH which is lower than the experimental values for Tl Mo Se and F 2 6 6 33 16 regime ; in contrast, our observed negative MR in Na Mo Se In Mo Se (13 and 17, respectively ) in spite of the smaller t in 2  d 6 6 2 6 6 > corresponds to a delocalization of gapless electronic states and Na Mo Se (Supplementary Fig. 2; see Methods for magnetic 2  d 6 6 provides additional evidence for a screened Coulomb interaction. field orientation details). This anisotropy is also far smaller than pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The MR switches sign again below T and becomes positive the measured conductivity ratio at 300 K: s =s ¼ 57. In pk == ? (Fig. 3e): as we shall now demonstrate, this is a signature of comparison, close agreement is obtained between the anisotropies superconductivity. in x and s for Li Mo O (ref. 39), where the effects of ==;? ==;? 0.9 6 17 disorder are believed to be weak . The disparate anisotropies in Na Mo Se arise from a strong suppression of x , thus 2  d 6 6 == illustrating the essential role of disorder in controlling the low- Superconducting transitions in Na Mo Se . The presence 2  d 6 6 15,16,35 temperature properties of Na Mo Se . of a superconducting ground state in Tl Mo Se and 2  d 6 6 2 6 6 In Mo Se implies that the peak in r(T) o6 K is likely to signify Although superconducting fluctuations are observed regardless 2 6 6 of the level of disorder in Na Mo Se , it is important to the onset of superconductivity in Na Mo Se . On cooling 2  d 6 6 2  d 6 6 crystals A–C in a dilution refrigerator, we uncover a identify whether phase-coherent long-range order develops in crystals D–F which exhibit super-critical disorder. In Fig. 4g–i, two-step superconducting transition characteristic of strongly 35–38 we demonstrate that r(T) in these samples still follows a 1D phase anisotropic q1D superconductors (Fig. 4a–c). Below T , pk slip model, albeit with a strongly enhanced contribution from superconducting fluctuations initially develop along individual quantum phase slips due to the increased disorder (Mo Se ) chains and r(T) is well-described by a 1D phase slip 6 6 N (Supplementary Note V). The fitting parameters for our 1D model (Supplementary Note V). Subsequently, a weak hump in phase slip analysis are listed in Supplementary Table II. A weak r(T) emerges (Fig. 4d–f) at temperatures ranging from B 0.95 K Meissner effect also develops in the magnetization below B 3.5 K (crystal A)toB 1.7 K (crystal C). This hump signifies the onset of in crystals D and E (Fig. 4g,h,j), but is rapidly suppressed by a transverse phase coherence due to inter-chain coupling. Cooper magnetic field. Low transverse phase stiffness is common in q1D pairs can now tunnel between the chains and a Meissner effect is superconductors: for example, bulk phase coherence in carbon expected to develop, but we are unable to observe this since 1.7 K lies below the operational range of our magnetometer. Analysis of nanotube arrays is quenched by 2–3 T, yet pairing persists up to 28 T . The superconducting volume fraction corresponding to the current–voltage characteristics indicates that a phase-coherent superconducting ground state is indeed established at low the magnitude of this Meissner effect is also unusually low: NATURE COMMUNICATIONS | 7:12262 | DOI: 10.1038/ncomms12262 | www.nature.com/naturecommunications 5 0.32 0.36 0.40 0.32 0.36 0.40 0.32 0.36 0.40 0.24 0.28 0.32 0.28 0.32 0.36 0.28 0.32 0.36 (Ωm) –1 (Sm ) –1 (Sm ) R(H) / R (%) max R(H) / R (%) max R(H) / R (%) max ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12262 a b c d e f 1E–4 1E–5 2E–5 6E–5 1E–5 8E–6 1E–5 4E–5 1E–5 6E–6 1E–6 1E–5 4E–6 2E–5 1E–6 1E–6 2E–6 1E–7 A B C A B C 0 0 0123 0123 0246 0.5 1.0 1.5 1.0 1.5 2.0 1.0 1.5 2.0 T (K) T (K) T (K) T (K) T (K) T (K) g j h i FC 0.1T 0.1 FC 0.1T 0 0 FC 0.1T 0.6 ZFC 2.0 ZFC 1.5 0.0 ZFC 0.5 –2 –2 FC 1.5 0.0025T –0.1 1.0 0.4 –4 –4 ZFC 1.0 –0.2 0.3 –6 –6 0.5 D E F E –0.3 0.5 2.0 2.5 0246 0246 0246 T (K) T (K) T (K) T (K) Figure 4 | Resistive and magnetic superconducting transitions in Na Mo Se . (a–c) Electrical resistivity r(To6 K) for crystals A–C. Coloured points 2  d 6 6 represent experimental data; black lines are fits to a 1D model incorporating thermal and quantum phase slips (Supplementary Note V). (d–f) Zoom views of r(T) in crystals A–C, plotted on a semi-logarithmic scale. The low-temperature limit of our 1D phase slip fits is signalled by a hump in r(T), highlighted by the transition from solid to dashed black fit lines: this corresponds to the onset of transverse phase coherence. In quasi-one-dimensional (q1D) superconductors, such humps form due to finite-size or current effects during dimensional crossover .(g–i) r(To6K) for the highly-disordered crystals D–F. Coloured points represent experimental data; black lines are fits to the same 1D phase slip model as in a–c, which accurately reproduces the broad superconducting transitions due to an increased quantum phase slip contribution (Supplementary Note V). Inhomogeneity and spatial fluctuations of the order parameter are expected to blur the characteristic hump in r(T) at dimensional crossover, thus explaining its absence from our data as the disorder rises. In g and h, we also plot zero-field-cooled/field-cooled (ZFC/FC) thermal hysteresis loops displaying the Meissner effect in the magnetic susceptibility w(T); j shows a zoom view of the susceptibility in crystal E. Data were acquired with the magnetic field parallel to the crystal c-axis and a paramagnetic background has been subtracted. The small diamagnetic susceptibilitiesjj w  1 are due to emergent pairing inhomogeneity creating isolated superconducting islands ; jj w is further decreased by the large magnetic penetration depth perpendicular to the c-axis in q1D crystals. o0.1%. Magnetic measurements of the superconducting volume Let us now examine the effects of disorder on the super- fraction in q1D materials invariably yield values o100%, since conducting ground state. Figure 5a illustrates T rising pk the magnetic penetration depth l normal to the 1D axis can monotonically from crystal A to F. Plotting T as a function of ab pk reach several microns and diamagnetic flux exclusion is r(300 K) (which is an approximate measure of the static disorder incomplete. For a typical Na Mo Se crystal of diameter in each crystal), we observe a step-like feature between crystals C 2  d 6 6 dB100mm, we estimate that a 0.1% volume fraction would and D, that is, at the critical disorder (Fig. 5b). Strikingly, the require l B10mm, which seems excessively large. Conversely, an characteristic VRH temperature T which we extract from our c 0 array of phase-fluctuating 1D superconducting filaments would r(T) fits (Fig. 3a) displays an identical dependence on r(300 K). not generate any Meissner effect at all. We therefore attribute the This implies that disorder controls both the superconducting unusually small Meissner signal to inhomogeneity in the ground state and the insulating tendency in r(T) at low superconducting order parameter, which is predicted temperature. The positive correlation between T and T pk 0 11,12,41,42 to emerge in the presence of intense disorder . (Fig. 5c) confirms that the onset temperature for superconducting In an inhomogeneous superconductor, Meissner screening is fluctuations (and hence the pairing energy D ) is enhanced by achieved via Josephson coupling between isolated super- localization in Na Mo Se . A concomitant increase in the 2  d 6 6 conducting islands . Within a single super-critically disordered transverse coherence temperature (Supplementary Note VI) Na Mo Se crystal, we therefore anticipate the formation implies that some enhancement in the phase stiffness also occurs. 2  d 6 6 of multiple Josephson-coupled networks comprising individual Super-critical disorder furthermore enables superconducting superconducting filaments. The total magnitude of the fluctuations to survive in high magnetic fields (Fig. 5d–g). In diamagnetic screening currents flowing percolatively through crystal C (which lies below the q1D mobility edge), super- each network will be much smaller than that in a homogeneous conductivity is completely quenched at all temperatures (that is, sample due to the smaller d/l ratio, thus diminishing the T -0) by H ¼ 4 T (Fig. 5d,f). A giant negative MR reappears for ab pk Meissner effect. H44 T (Fig. 3e), confirming that superconductivity originates from pairing between localized electrons. In contrast, the peak at r(T ) in the highly-disordered crystal F is strikingly resistant to pk magnetic fields (Fig. 5e,g): at T ¼ 4.6 K, our observed H ¼ 14 T, Enhancement of superconductivity by disorder. We have c2 established a clear influence of disorder on electrical transport in which exceeds the weak-coupling Pauli pair-breaking limit H ¼ 3 T by a factor 44 (see Supplementary Note VII for a Na Mo Se (Figs 2 and 3) and demonstrated that the peak in 2  d 6 6 P r(T)at T corresponds to the onset of superconductivity (Fig. 4). derivation of H (T)). A similar resilience is evident from the pk 6 NATURE COMMUNICATIONS | 7:12262 | DOI: 10.1038/ncomms12262 | www.nature.com/naturecommunications (Ωm) –3 Dimensionless susceptibility  ×10 (Ωm) (Ωm) –3 Dimensionless susceptibility  ×10 (Ωm) (Ωm) (Ωm) (Ωm) (Ωm) –3 Dimensionless susceptibility  ×10 (Ωm) NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12262 ARTICLE a b c 6 6 A B C D E F 1.0 1E4 0.9 1E3 0.8 1E2 23456 1E–7 1E–6 1E–5 1E2 1E3 1E4 T (K)  (300K) (Ωm) T (K) d e f C F C F 2.5 1.2E–4 4 T 14 T 2.0 1E–4 2 T 12 T 1 T 8 T 8E–5 0 T 1.5 0 T 6E–5 1.0 4E–5 0.5 2E–5 0 0.0 0 5 10 15 0 5 10 15 123 4.5 5.0 5.5 T (K) T (K) T (K) T (K) Figure 5 | Disorder controls the divergent electrical resistivity and enhances superconductivity. (a) Zoom view of the temperature-dependent electrical resistivity r(T) at the onset of superconductivity in all crystals, normalized to r(T ). (b) Evolution of the characteristic variable range hopping temperature pk T and the superconducting onset temperature T with r(300 K). The step at 10 O m corresponds to the critical disorder, that is, the quasi-one- 0 pk dimensional mobility edge. Error bars in r(300 K) are determined from the experimental noise level and our measurement resolution for the crystal dimensions. The error in T corresponds to its s.d., obtained from our variable range hopping fitting routine. (c) T versus T for each crystal, confirming 0 pk 0 the positive correlation between superconductivity and disorder. Data from three additional crystals which broke early during our series of measurements (Supplementary Note III) are also included (black circles). (d,e) Suppression of superconductivity with magnetic field H perpendicular to the c-axis for crystals C (d) and F (e). (f,g) Upper critical field H (T), equivalent to T (H), for crystals C (f) and F (g). Error bars in H (T) correspond to the error in c2 pk c2 determining the maximum in r(T,H). @H ðTÞ=@Tj ¼ 5:1TK and 24 T K for C and F, respectively. c2 pk positive MR in crystal D, which persists up to at least 14 T at 1.8 K that the localization length x remains larger than the coherence (Fig. 3e). Triplet pairing is unlikely to occur in Na Mo Se length (that is, the Cooper pair radius). However, experiments 2  d 6 6 (since scattering would rapidly suppress a nodal order parameter) have invariably shown superconductivity to be destroyed and orbital limiting is also suppressed (since vortices cannot form by disorder, due to enhanced Coulomb pair-breaking , phase 42,46,47 10,48 across phase-incoherent filaments). Our data therefore suggest fluctuations or emergent spatial inhomogeneity . that disorder lifts H , creating anomalously strong correlations In particular, increasing disorder in Li Mo O (one of the P 0.9 6 17 which raise the pairing energy D (refs 10,11) above the weak- few q1D superconductors extensively studied in the literature) coupling 1.76 k T . A direct spectroscopic technique would be monotonically suppresses superconductivity . Therefore, the key B pk required to determine the absolute enhancement of D , since question arising from our work is why the onset temperature for spin-orbit scattering from the heavy Mo ions will also contribute superconductivity rises with disorder in Na Mo Se ,in 2  d 6 6 to raising H . contrast to all other known materials? Disorder acts to enhance the matrix element for e –e interactions. This may be explained qualitatively by considering Discussion that all conduction electron wavefunctions experience the same disorder-induced potential, developing inhomogeneous multi- The emergence of a superconducting ground state in Na Mo Se places further constraints on the origin of the fractal probability densities and hence becoming spatially 2  d 6 6 correlated. Such enhanced correlations have been predicted to normal-state divergence in r(T). Our electronic structure calculations indicate that the q1D Fermi surface of Na Mo Se increase the Cooper pairing energy : in the absence of pair- 2  d 6 6 breaking by long-ranged Coulomb interactions, this will lead to a is almost perfectly nested: any incipient electronic DW would 11–13,51,52 therefore gap the entire Fermi surface, creating clear signatures of rise in the superconducting transition temperature . A proposal to observe this effect in superconducting hetero- a gap in r(T) and leaving no electrons at E to form a superconducting condensate. In contrast, our VRH fits and MR structures with built-in Coulomb screening (by depositing superconducting thin films on substrates with high dielectric data do not support the formation of a DW gap, and a superconducting transition occurs at low temperature. Electrons constants) has not yet been experimentally realised. However, our VRH dimensionality d41 (Fig. 3a) and negative MR (Fig. 3d,e) must therefore remain at E for all T4T , indicating that r(T) F pk diverges due to disorder-induced localization rather than any both point towards a weak or screened Coulomb repulsion, while other insulating instability. the power laws and broad minima in r(T) at high temperature (Fig. 2a) indicate a Luttinger parameter K 41. These results all It has been known since the 1950s that an s-wave super- 44,45 conducting order parameter is resilient to disorder , provided imply that e –e interactions in Na Mo Se are attractive. 2  d 6 6 NATURE COMMUNICATIONS | 7:12262 | DOI: 10.1038/ncomms12262 | www.nature.com/naturecommunications 7 /(T ) (Ωm) pk (Ωm) T (K) H (T) c2 T (K) pk T (K) pk H (T) c2 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12262 (For comparison, K B0.25 in Li Mo O and the e –e Beyond enhancing superconductivity, the ability to simulta- r 0.9 6 17 7,8 interaction is repulsive .) Phonon-mediated coupling—the neously modulate band filling, disorder and dimensionality Cooper channel—therefore appears to dominate over the promises a high level of control over emergent order, including Coulomb repulsion in Na Mo Se , suggesting that the usual DWs and magnetic phases. More generally, Na Mo Se 2  d 6 6 2  d 6 6 disorder-induced Coulomb pair-breaking may be avoided. Below and other similar interrupted strand materials may be ideal the q1D mobility edge, our rise in T is quantitatively compatible environments in which to study the evolution of many-body pk with a weak multifractal scenario (Supplementary Fig. 8, electron localization beyond the non-interacting Anderson limit. Supplementary Note VIII), providing a possible explanation for the enhancement of superconductivity which merits further Methods theoretical attention. Crystal growth and initial characterization. A series of Na Mo Se crystals 2  d 6 6 The fact that no experimental examples of q1D materials with was grown using a solid-state synthesis procedure. The precursor materials were MoSe , InSe, Mo and NaCl, all in powder form. Before use, the Mo powder was attractive e –e interactions have yet been reported poses the 2 reduced under H gas flowing at 1,000 C for 10 h, to eliminate any trace of oxygen. question why Na Mo Se should be different. Although strong 2  d 6 6 The MoSe was prepared by reacting Se with H -reduced Mo in a ratio 2:1 inside a 2 2 electron–phonon coupling is known to play an important role in purged, evacuated and flame-baked silica tube (with a residual pressure of 16,53 the physics of molybdenum cluster compounds , we propose B10 mbar argon), which was then heated to B700 C for 2 days. InSe was synthesized from elemental In and Se in an evacuated sealed silica tube at 800 C that the disordered q1D nature of Na Mo Se may instead play 2  d 6 6 for 1 day. Powder samples of Na Mo Se were prepared in two steps. First, 2  d 6 6 the dominant role, by suppressing the Coulomb repulsion. In the In Mo Se was synthesized from a stoichiometric mixture of InSe, MoSe and Mo, 2 6 6 2 presence of disorder, a q1D material can be regarded as a parallel heated to 1,000C in an evacuated sealed silica tube for 36 h. Second, an ion array of ‘interrupted strands’ , that is, a bundle of finite-length exchange reaction of In Mo Se with NaCl was performed at 800 C, using a 10% 2 6 6 nanowires. The electric polarizability of metallic nanoparticles is NaCl excess to ensure total exchange as described in ref. 61. All starting reagents were found to be monophase on the basis of their powder XRD patterns, acquired strongly enhanced relative to bulk materials , although this effect using a D8 Bruker Advance diffractometer equipped with a LynxEye detector is usually cancelled out by self-depolarization. The geometric (CuKa radiation). Furthermore, to avoid any contamination by oxygen and depolarization factor vanishes for q1D symmetry, leading to giant moisture, the starting reagents were kept and handled in a purified argon-filled dielectric constants e which rise as the filament length increases . glovebox. To synthesize single crystals, a Na Mo Se powder sample (of mass B5g) This effect was recently observed in Au nanowires , with e 2  d 6 6 7 was cold-pressed and loaded into a molybdenum crucible, which had previously reaching 10 .InNa Mo Se , we therefore anticipate that 2  d 6 6  5 been outgassed at 1,500 C for 15 min under a dynamic vacuum of B10 mbar. the long-range Coulomb repulsion in an individual (Mo Se ) 6 6 l The Mo crucible was subsequently sealed under a low argon pressure using an filament (loN) will be efficiently screened by neighbouring arc-welding system. The Na Mo Se powder charge was heated at a rate of 2  d 6 6 300 Ch up to 1,750 C, held at this temperature for 3 h, then cooled at filaments . This intrinsic screening provides a natural 100 Ch down to 1,000 C and finally cooled naturally to room temperature explanation for attractive e –e interactions and suppresses within the furnace. Crystals obtained using this procedure have a needle-like shape Coulomb pair-breaking in the superconducting phase. with length up to 4 mm and a hexagonal cross-section with typical diameter It has been suggested that impurities can increase the r150mm. Initial semi-quantitative microanalyses using a JEOL JSM 6400 scanning electron microscope equipped with an Oxford INCA EDX spectrometer indicated temperature at which transverse phase coherence is established that the Na contents ranged between 1.7 and 2, that is, up to 15% deficiency. The in q1D superconductors . This effect cannot be responsible for Na deficiency results from the high temperatures used during the crystal growth our observed rise in T , which corresponds to the onset of 1D pk process coupled with the small size of the Na ion: it cannot be accurately controlled superconducting fluctuations on individual (Mo Se ) filaments. 6 6 l within the conditions necessary for crystal growth. Since In Mo Se is known to be superconducting below 2.85 K , it is important We also point out that the finite-size effects which influence 2 6 6 59 60 to consider the possibility of In contamination in our samples. The Na/In ion critical temperatures in granular or nanomaterials are not 61,62 exchange technique used during synthesis is known to be highly efficient and relevant in Na Mo Se : quantum confinement is absent in 2  d 6 6 In Mo Se decomposes above 1,300 C, well below our crystal growth temperature 2 6 6 homogeneously-disordered crystalline superconductors and (1,750 C). This precludes the presence of any superconducting In Mo Se 2 6 6 hence no peaks form in N(E ). These mechanisms are discussed (or In-rich (In,Na) Mo Se ) filaments in our crystals. Diffuse X-ray scattering F 2 6 6 measurements accordingly reveal none of the Huang scattering or disk-like Bragg in detail in Supplementary Note IX. reflections which would be produced by such filaments. Furthermore, EDX In summary, we have presented experimental evidence for the spectrometry is unable to detect any In content in our crystals, while inductively- enhancement of superconductivity by disorder in Na Mo Se . 2  d 6 6 coupled plasma mass spectrometry indicates a typical In residual of o0.01%, that The combination of q1D crystal symmetry (and the associated is, o0.0002 In atoms per unit cell. The electronic properties of Na Mo Se 2  d 6 6 crystals will remain unaffected by such a tiny In residual in solid solution. dimensional crossover), disorder and incommensurate band filling in this material poses a challenge to existing 1D/q1D theoretical models. Although the normal-state electrical resistivity Electrical transport measurements. Before all measurements, the as-grown of Na Mo Se is compatible with theories for disordered 1D crystal surfaces were briefly cleaned with dilute hydrochloric acid (to remove any 2  d 6 6 residue from the Mo crucible and hence minimize the contact resistance), followed systems with attractive e –e interactions, we establish several by distilled water, acetone and ethanol. Four Au contact pads were sputtered onto unusual low-temperature transport properties which deserve the upper surface and sides of each crystal using an Al foil mask; 50mm Au wires future attention. These include a resistivity which diverges were then glued to these pads using silver-loaded epoxy cured at 70 C (Epotek following a q1D VRH law for all levels of disorder, the existence E4110). Special care was taken to thoroughly coat each end of the crystal with epoxy, to ensure that the measurement current passed through the entire crystal. of a critical disorder or q1D mobility edge where T ET , and a min x 2 All contacts were verified to be Ohmic at room temperature before and after each strongly frequency-dependent conductivity s(o)Bo in crystals series of transport measurements, and at T ¼ 4 K after cooling. Typical contact with super-critical disorder. At temperature T , 1D super- pk resistances were of the order of 2O at 300 K. The transverse conductivity s was conducting fluctuations develop, and a phase-coherent ground estimated at room temperature using a four-probe technique, with contacts on opposite hexagonal faces of a single crystal. The temperature dependence of the state is established via coupling between 1D filaments at lower transverse resistivity r (T) has never been accurately measured in M Mo Se due > 2 6 6 temperature. As the disorder rises, T increases: in our most- pk to the exceptionally large anisotropies, small crystal diameters and high fragility, disordered crystals, the survival of superconducting fluctuations 15 even in the least anisotropic Tl Mo Se which forms the largest crystals . 2 6 6 in magnetic fields at least four times larger than the Pauli limit Low-frequency four-wire ac conductivity measurements were performed in two separate cryogen-free systems: a variable temperature cryostat and a dilution suggests that the pairing energy may be unusually large. refrigerator, both of which may be used in conjunction with a superconducting We conclude that deliberately introducing disorder into q1D vector magnet. The ac conductivity was measured using a Keithley 6100 current crystals represents a new path towards engineering correlated source, a Stanford SRS850 lock-in amplifier with input impedance 10 MO and (for electron materials, in remarkable contrast with the conventional low resistances, that is, weakly-disordered samples) a Stanford SR550 preamplifier blend of strong Coulomb repulsion and a high density of states. with input impedance 100 MO. Data from several crystals were cross-checked using 8 NATURE COMMUNICATIONS | 7:12262 | DOI: 10.1038/ncomms12262 | www.nature.com/naturecommunications NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12262 ARTICLE a Quantum Design Physical Property Measurement System with the standard 19. Narduzzo, A., Enayati-Rad, A., Horii, S. & Hussey, N. E. Possible coexistence of inbuilt ac transport hardware: both methods generate identical, reproducible data. local itinerancy and global localization in a quasi-one-dimensional conductor. 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Commun. 7:12262 doi: 10.1038/ncomms12262 (2016). one-monodimensional ternary molybdenum chalcogenides M Mo X 2 6 6 (X ¼ Se,Te; M ¼ Li,Na..Cs). Mater. Res. Bull. 19, 915–924 (1984). 63. Lepetit, R., Monceau, P., Potel, M., Gougeon, P. & Sergent, M. This work is licensed under a Creative Commons Attribution 4.0 Superconductivity of the linear chain compound Tl Mo Se . J. Low Temp. Phys. International License. The images or other third party material in this 2 6 6 56, 219–235 (1984). article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. Acknowledgements To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ We thank Alexei Bosak (Beamline ID28, ESRF Grenoble) for assistance with data collection and processing, and Igor Burmistrov, Vladimir Kravtsov, Tomi Ohtsuki and Vincent Sacksteder IV for stimulating discussions. The Swiss-Norwegian Beamlines r The Author(s) 2016 10 NATURE COMMUNICATIONS | 7:12262 | DOI: 10.1038/ncomms12262 | www.nature.com/naturecommunications

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