Phonon Mode Spectroscopy, Electron-Phonon Coupling and the Metal-InsulatorTransition in Quasi-One-Dimensional M2Mo6Se6

A.P. Petrovic,∗ R. Lortz, G. Santi, M. Decroux, H. Monnard, and Ø. FischerDPMC-MaNEP, Universite de Geneve, Quai Ernest-Ansermet 24, 1211 Geneve 4, Switzerland

L. Boeri and O.K. AndersenMax Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany

J. KortusInstitut fur Theoretische Physik, TU Bergakademie Freiberg,

Leipziger Strasse 23, D-09596 Freiberg, Germany

D. Salloum, P. Gougeon, and M. PotelSciences Chimiques, CSM UMR CNRS 6226, Universite de Rennes 1,

Avenue du General Leclerc, 35042 Rennes Cedex, France(Dated: November 2, 2018)

We present electronic structure calculations, electrical resistivity data and the first specific heatmeasurements in the normal and superconducting states of quasi-one-dimensional M2Mo6Se6 (M= Tl, In, Rb). Rb2Mo6Se6 undergoes a metal-insulator transition at ∼ 170 K: electronic structurecalculations indicate that this is likely to be driven by the formation of a dynamical charge densitywave. However, Tl2Mo6Se6 and In2Mo6Se6 remain metallic down to low temperature, with super-conducting transitions at Tc = 4.2 K and 2.85 K respectively. The absence of any metal-insulatortransition in these materials is due to a larger in-plane bandwidth, leading to increased inter-chainhopping which suppresses the density wave instability. Electronic heat capacity data for the su-perconducting compounds reveal an exceptionally low density of states DEF = 0.055 states eV−1

atom−1, with BCS fits showing 2∆/kBTc ≥ 5 for Tl2Mo6Se6 and 3.5 for In2Mo6Se6. Modellingthe lattice specific heat with a set of Einstein modes, we obtain the approximate phonon densityof states F (ω). Deconvolving the resistivity for the two superconductors then yields their electron-phonon transport coupling function α2

trF (ω). In Tl2Mo6Se6 and In2Mo6Se6, F (ω) is dominated byan optical “guest ion” mode at ∼ 5 meV and a set of acoustic modes from ∼ 10-30 meV. Rb2Mo6Se6exhibits a similar spectrum; however, the optical phonon has a lower intensity and is shifted to ∼8 meV. Electrons in Tl2Mo6Se6 couple strongly to both sets of modes, whereas In2Mo6Se6 onlydisplays significant coupling in the 10-18 meV range. Although pairing is clearly not mediated bythe guest ion phonon, we believe it has a beneficial effect on superconductivity in Tl2Mo6Se6, givenits extraordinarily large coupling strength and higher Tc compared to In2Mo6Se6.

PACS numbers: 71.30.+h, 74.25.-q, 74.70.Dd

I. INTRODUCTION

The M2Mo6Se6 (M = Tl, In, Rb, Li, Na, K, Cs) sys-tem was first discovered by Potel et al.1 and is closelyrelated to the well-known quasi-three-dimensional (quasi-3D) Chevrel Phase compounds. Rather than comprisingindividual “zero-dimensional” Mo6X8 (X = S, Se, Te) oc-tahedral clusters coupled by a metallic cation, these ma-terials are composed of quasi-1D (Mo6Se6)∞ chains ori-ented along the z axis, weakly coupled by M ions. OnlyTl2Mo6Se6 and In2Mo6Se6 are superconducting, with Tc

= 3 - 6.5 K (varying between samples) and ∼ 2.9 K re-spectively. In contrast, Rb2Mo6Se6 undergoes a broadmetal-insulator transition between 100 K and 200 K.2

Little data currently exists in the literature for the re-maining members of the family, although it is knownthat they exhibit similar metal-insulator transitions anddo not become superconducting under ambient pressureat low temperature.3,4

Reduced dimensionality and its effect on supercon-

ductivity remains one of the central issues in contem-porary condensed matter physics research. Since thelate 1970s, numerous unconventional superconductorsdisplaying highly anisotropic properties in both the nor-mal and superconducting states have been discovered.Among these, notable examples include the quasi-2Dhigh-temperature cuprate superconductors (HTS) andthe quasi-1D organic Bechgaard salts. However, themost strongly 1D superconductors synthesized to datehave attracted remarkably little attention over the years.Tl2Mo6Se6 and In2Mo6Se6 boast anisotropy ratios ε =

H‖c2/H

⊥c2 ≥ 12.0 and 17.2 respectively,5–7 significantly

greater than ε ≈ 8.5 in (TMTSF)2ClO4.8 Furthermore,these materials do not possess any intrinsic magnetism,thus rendering them an ideal uncomplicated system forthe study of low-dimensional superconductivity.

In comparison with the HTS, the number of publi-cations existing for M2Mo6Se6 is around three ordersof magnitude smaller. Early work concentrated on theelectrical transport6,9 and magnetic properties,10 imme-

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diately revealing large anisotropies in both the normal-state resistivity and superconducting coherence lengthfor Tl2Mo6Se6. Two distinct classes of Tl2Mo6Se6 wereidentified6 by the behaviour of their longitudinal resistiv-ity: A-type samples with conventional metallic behaviourdown to low temperature or B-type samples displayinga broad minimum for T < ∼ 80 K followed by an up-turn reminiscent of charge density wave (CDW) forma-tion. More recent measurements5 have shown that thecoherence length perpendicular to the chain axis ξ⊥ isat most 75 A, a value not significantly larger than thatfound in some HTS. In an important parallel with organicquasi-1D superconductors,11 the upper critical field per-pendicular to the chain axis z, H⊥c2, does not saturatedown to 50 mK.12 Hall effect measurements by the sameauthors display a regime crossover at T ≈ 80 K whichthey attribute to the onset of a CDW or spin densitywave (SDW). However, there is no support for the forma-tion of a CDW in normal-state resistivity curves for theA-type samples, which nonetheless display a Hall effectcrossover. Furthermore, the weak temperature-invariantdiamagnetism in Tl2Mo6Se6 revealed by ac susceptibil-ity measurements2 does not encourage a SDW interpre-tation.

The discovery that the application of uniaxial stressalong the z axis in Tl2Mo6Se6 suppresses superconduc-tivity and induces a metal-insulator transition increasedthe evidence for this system being close to a CDW insta-bility, in particular due to the non-linear I−V curves andbroadband noise characteristic of density wave motionobserved.13 Conversely, hydrostatic pressure increasesthe conductivity of Tl2Mo6Se6 in the normal state butstill suppresses superconductivity.14

At first glance, the quasi-one-dimensional nature of theM2Mo6Se6 family renders them strong candidates to un-dergo a Peierls transition, so it was initially a mysteryas to why Tl2Mo6Se6 and In2Mo6Se6 remained metallicdown to low temperature. Early attempts to resolve thisquestion focussed on the calculated band structure15,16

which displayed three contributions to the Fermi sur-face: a broad singly-degenerate quasi-1D Mo d “helix”band and two doubly-degenerate 3D “octahedral” elec-tron pockets at the zone boundary. The occupancy ofthese 3D pockets was believed to stabilise the structureagainst a Peierls transition and the authors of all ensuingpublications attempted to interpret their results withinthis multi-band framework. However, band structure cal-culation techniques have significantly advanced since thenon-self-consistent approaches of the early 1980s. It istherefore instructive to recalculate the band structureof M2Mo6Se6 using a fully self-consistent method andcompare our results with the existing calculations forTl2Mo6Se6.

The nature of the superconducting state in Tl2Mo6Se6and In2Mo6Se6 also remains very unclear, particularlysince recent scanning tunnelling microscopy (STM) ex-periments17 on the related quasi-3D Chevrel Phase su-perconductor PbMo6S8 have provided strong evidence

for a highly anisotropic or noded gap function. In-elastic neutron scattering measurements of the phonondensity of states (PDoS)18 revealed a strong low-energyEinstein-like optical mode attributed to vibrations of theM “guest” ions between the Mo6Se6 chains, as well ashigher energy intra-chain modes similar to those seen inthe 3D Chevrel phases.19 This finding is supported byearly normal-state specific heat data20 which unfortu-nately lacked sufficient resolution to provide any infor-mation on the superconducting state. However, no stud-ies of the electron-phonon coupling were performed; norhave any tunnelling experiments been carried out on theM2Mo6Se6 system.

Motivated by recent discoveries in boride21,22 and β-pyrochlore23 systems where superconductivity is medi-ated by a low-energy rattling phonon, we therefore de-cided to measure the PDoS and electron-phonon couplingfor Tl2Mo6Se6, In2Mo6Se6 and Rb2Mo6Se6 by decon-volving normal-state specific heat and resistivity data.These measurements are analysed in parallel with ourspecific heat and resistivity data below Tc for M = Tl,Inand in the insulating phase for M = Rb. In addition,we have performed a complete theoretical analysis of theelectronic structure and parameters governing the metal-insulator transition in M2Mo6Se6. By combining our ex-perimental data with the trends indicated by our newband structures, we hope to remove some of the con-fusion surrounding the mechanism for superconductivityand the effects of low-dimensionality in this fascinatingsystem on the borderline between superconducting andinsulating instabilities.

II. THEORY

A. Crystal Structure

The crystal structure of the compounds with chemicalformula M2Mo6Se6 is shown in Fig. 1, viewed both par-allel and perpendicular to the z axis. Mo and Se atomsform quasi-1D (Mo6Se6)∞ chains oriented along the zaxis, separated by M ions in a zig-zag formation. Thechains consist of inner Mo and outer Se triangles, stackedwith a c/2 separation along the z axis and rotated 180

with respect to each other. The axes of the Mo-Se trian-gles are aligned with each other, but rotated by 10 fromthe lattice vectors in the x − y plane. We may considerthe chains to be a linear condensation of Mo6S8 clustersvia face-sharing of the Mo6 octahedra. These clustersare the building blocks of the related quasi-3D Chevrelphases.

The conventional unit cell is hexagonal (space groupP63/m), and contains two formula units (f.u.); M atomsoccupy 2d positions while Mo and Se atoms occupy 6hpositions. The hexagonal lattice parameters (determinedby X-ray diffraction) are aH = 8.854, 8.934, 9.257 A andcH = 4.493, 4.494, and 4.487 A respectively for In, Tland Rb-based crystals. It can immediately be seen that

3

FIG. 1: Crystal structure of M2Mo6Se6 compounds in the001 plane (above) and side view (below) of the (Mo6Se6)∞chains. Darker (lighter) symbols indicate atoms sitting oneven (odd) planes respectively. The local coordinate systemused to plot the partial Mo, Se characters in Fig. 2 is alsoshown.

M xMo yMo xSe ySe

In 0.189 0.156 0.068 0.369Tl 0.187 0.154 0.067 0.366Rb 0.181 0.149 0.064 0.355

TABLE I: Optimized internal coordinates (Wyckoff positions)for Mo and Se in the three M2Mo6Se6 compounds consideredin this work (M=Tl,In,Rb).

the inter-chain distance correlates with the atomic radiusof the M atom, whereas cH (and hence the intra-chainatomic separation) remains roughly constant regardlessof M . Furthermore, the intra-chain atomic separationsare much smaller than the inter-chain distances. For ex-ample, in Tl the shortest Mo-Mo distance (for two Moatoms in the same triangle) is 2.66 A; the shortest Mo-Sedistance is 2.695 A and the Se-Se separation is 3.767 A.In contrast, the Mo-Mo inter-chain separation is 6.34 A.

The crystal structure thus provides an immediate in-dication of the strong anisotropy present in this familyof materials. In the following, we will show that a large

anisotropy is also found in the electronic structure anddiscuss its consequences for the physical properties ofM2Mo6Se6 compounds.

B. Electronic Structure

We have performed ab − initio Density FunctionalTheory (DFT) calculations of the electronic propertiesof M2Mo6Se6, for M =Tl, In, Rb, employing the thefull-potential Linear Augmented Plane Wave (LAPW)method.24,25 For all systems, we used the experimen-tal lattice parameters and optimized the internal coor-dinates: these optimized values are given in table I.

For Tl2Mo6Se6, where the experimental Wyckoff posi-tions are known,26 the optimized coordinates agree withthe experimental ones to better than 1 %; for In2Mo6Se6and Rb2Mo6Se6 we could not compare with experimen-tal data. Structural optimization yielded intra-chain dis-tances which do not depend (within computational accu-racy) on the nature of the M atom, whereas interchaindistances increase by 5 % going from Tl,In to Rb due tothe larger in-plane lattice constant.

FIG. 2: Band structure and partial DoSs of Tl2Mo6Se6; en-ergies are in eV and measured w.r.t. the Fermi level, DoSare in states/(eV cell). Colors indicate different l charac-ters (green=s, orange=p, red=d). Shaded areas, delimited bydashed lines, refer to the partial characters (Tl pz, Mo dxz,Se px) which provide the largest contribution at EF . For adefinition of the axes and atoms, see Fig. 1. An enlargementof the DoS around the Fermi level is also shown; the numbersindicate the enlargement factor.

Since the major features of the electronic structureof M2Mo6Se6 are the same for M=Tl, In, Rb, we willfirst present the electronic structure of Tl2Mo6Se6 andthen discuss the differences with In and Rb. Our calcu-lated electronic structure reproduces the main featuresof the earlier non-self-consistent calculations,15,16 exceptfor one crucial difference in the position of the Fermilevel. The electronic bands and partial densities of states

4

(DoS) are shown in Fig. 2, with all energies measuredrelative to the Fermi level. Se s and Tl d states formweakly-dispersing bands, located 15 eV and 12 eV belowthe Fermi level respectively. The Tl 6s states lie approx-imately 7 eV below the Fermi level; Tl is therefore ina nominal +1 valence state, effectively behaving like analkali metal. The bands lying between -5 and +5 eV aremainly of Mo d and Se p character, although there is asignificant hybridization with Tl p states as can be seenin Fig. 2. The Mo sp bands are displaced to higher ener-gies by covalent hybridisation with Se p bands and hencelie above +5 eV.

The 18 Se p bands are centered around -4 eV and over-lap with the lower portion of the 30 Mo d bands, whichextend over ±5 eV around the Fermi level. These aredivided into 12 bonding and 16 antibonding states, sep-arated by a pseudogap ∼ 1 eV wide around the Fermilevel. Two highly 1D bands cross the gap: they de-rive from the zone-folding of a single helix band. TheFermi level cuts the band structure at kz=π/c, exactlyat half filling. The corresponding Fermi Surface is shownin Fig. 3. In contrast with earlier calculations, we do notfind the additional small 3D pockets at the Fermi levelwhich were previously alleged to be responsible for thestability of the chains against any Peierls distortion andensuing density wave transition.15 The octahedron band(of Mo d 3z2 − r2 and x2 − y2 character) responsible forthese pockets in fact lies ∼ 0.1 eV above EF . Fig. 4shows that In2Mo6Se6 and Rb2Mo6Se6 have a very sim-ilar band structure to Tl2Mo6Se6, i.e. there are no otherbands at EF except for the helix band. Small variationsin the dispersion of this band must therefore control thestability of the M2Mo6Se6 family.

Let us now consider the relative anisotropy of theM2Mo6Se6 series, in terms of details of the dispersionof the helix band, using a simplified tight-binding modelwith large out-of-plane (W ) and small in-plane (w) band-widths. In the three rightmost panels of Fig. 2, we high-light the partial orbital characters which give the largestcontribution to the helix band: the dominant contribu-tion is from Mo dxz states. (The labels of the orbitalsrefer to the local coordinate systems centered on the Moatoms, shown in Fig. 1).

In fact, the Bloch states that form this band are builtfrom the in-phase linear combination of the three equiva-lent Mo xz orbitals which sit on each Mo3 triangle. Dueto the presence of a two-fold screw axis along the cen-ter of the chain, they form right and left-handed heliceswhich wind up along the chain axis. (For a definition ofthe axes, see Fig. 1). The out-of-plane bandwidth W isthus given by the hopping of one Mo dxz to another Modxz orbital on the next plane; the in-plane bandwidth isgiven by the hopping between the chains, mediated by Sepx and Tl(In) pz or Rb s orbitals (illustrated by yellowand green lines respectively in Fig. 2).

Due to symmetry, hopping via Tl(In) pz orbitals ismore effective than via Rb s orbitals. In Fig. 4 we showa close-up of the band structures ofM2Mo6Se6, decorated

Tl In RbW 7 7 8w 0.17 0.14 0.02λc 0.11 0.11 0.07D 6.74 6.74 7.70ω0,1 22 22 22ω0,2 27 27 27λ(ω0,1) 0.10 0.10 0.13λ(ω0,2) 0.07 0.07 0.09δ0 0 0 0.17ω1 6.6 6.6 11.2ω2 8.2 8.2 11.4

TABLE II: Parameters governing the metal-insulator transi-tion in M2Mo6Se6 (M = Tl,In,Rb) from local density approxi-mation (LDA) calculations. W and w are the out-of-plane andin-plane bandwidths of the helix band in eV: they determinethe dimensionless critical coupling constant λc (see Eq. 2).ω0,i are the bare frequencies of the two Peierls modes in meV.λ(ω0,i) are the corresponding dimensionless bare electron-phonon coupling constants (see Eq. 1); 2 δ0 is the value ofthe Peierls gap (Eq. 4) in eV. The renormalized phonon fre-quencies ωi (in meV) are obtained from Eqs. 3 and 5 for Tl,In and Rb respectively.

FIG. 3: Fermi surface of Tl2Mo6Se6, in a vertical plane cut-ting through the center of the hexagonal Brillouin zone, shownin Fig. 1.

with the partial characters associated with the M atoms.Here, we observe that the contribution of Tl and In pzstates to the helix band is much stronger than that of theRb s states. This agrees well with the significant In p or-bital contribution to the conduction band in In2Mo6Se6suggested by nuclear magnetic resonance spectroscopy.27

Correspondingly, the in-plane bandwidth is reduced by afactor of ∼ 10 in Rb2Mo6Se6. From a fit of these threeband structures, we extract the values of W and w re-ported in table II.

5

FIG. 4: Band structures of M2Mo6Se6, for M=Tl,In andRb decorated with partial M characters. Energies, measuredw.r.t. the Fermi level, are in eV.

C. Mechanisms for Metal-Insulator Transitions

The first possibility for the metal-insulator transitionis a CDW instability, due to the interaction of the he-lix band with phonons (i.e. a Peierls transition). InM2Mo6Se6, there are two Peierls-like modes which couldlead to dimerization of the chains and open a gap at EF :the eigenvectors are such that the Mo triangular unitson subsequent planes move out-of-phase with respect toeach other and the surrounding Se triangles can then bedisplaced either in or out of phase.

Examining these two modes, we use local density ap-proximations (LDA) to determine two bare frequenciesω0,1 and ω0,2 which we have found to be independent ofthe nature of the M guest ion. Our calculated values areω0,1 = 22 meV and ω0,2 = 27 meV. These frequencies in-clude the response of the whole electronic band structureto the phonon perturbations, with the exception of thehelix band, which we calculate analytically below. Thisprocedure is more accurate than a direct LDA calcula-tion of the phonon frequency, which becomes extremelyinaccurate as the in-plane bandwidth w of the helix bandtends to zero.

The effect of both modes on the band structure isthe same: a frozen displacement eu of the atoms alongthe renormalized eigenvector opens a gap 2δ = 2Du atthe Fermi wavevector kz = π/c. This defines the de-formation potential D which (together with the barephonon frequencies ω0,i and the DoS at the Fermi levelN(0) = 1/W ) determines the bare electron phonon cou-pling constants λ(ω0,i) through the relation:

λ(ω0,i) =D2

WMω20,i

. (1)

Due to the finite in-plane bandwidth of the helix band, aPeierls transition can take place only if this bare electron-phonon coupling constant λ(ω0,i) exceeds a critical valueλc:

λc =1

2 (ln(W/w) + 1). (2)

If λ(ω0,i) < λc, there is no Peierls transition, but thebare phonon frequency is renormalized:

ωi = ω0,1

√(1− λ(ω0,i)

λc

)(3)

If λ(ω0,i) > λc, there is a Peierls transition, with a gap2δ0 = 2Du0:

2δ0 = 2W exp(− 1

2λ(ω0,i)), (4)

The frequency of the ith phonon in the off-centre posi-tion is then given by:

ωi = ω0,i

√2λ(ω0,i). (5)

Table II lists the relevant LDA parameters forM=Tl,In,Rb. It may readily be seen that all M2Mo6Se6compounds are close to the CDW instability, since thevalues of the critical interaction parameters λc are verylow compared to the bare electron-phonon coupling pa-rameters λ0,i.

For Tl2Mo6Se6 and In2Mo6Se6, the bare electron-phonon coupling constants for both the lower andthe higher Peierls modes are slightly below λc; forRb2Mo6Se6 λ(ω0,1) is well above the critical value, thusindicating that a Peierls transition is likely to occur witha gap 2δ = 0.34 eV. However, due to the small value ofλ(ω0,1), the off-center minimum of the phonon potentialis very shallow, implying that the transition is of dynam-ical rather than static character. This means that thedensity wave formation may not be characterised by astatic structural modulation, as is the case for “classi-cal” CDW systems such as NbSe2.

An alternative explanation for the observed metal in-sulator transition in M2Mo6Se6 compounds could be aspin density wave (SDW) instability. A staggered an-tiferromagnetic order of Mo3 units is the spin analogueof the Peierls distortion. The spin density wave (SDW)transition is regulated by the magnetic coupling constantλI= 1

4I3

1W , where the Stoner parameter I can be esti-

mated from the atomic value for Mo: 0.6 eV. λI is oneorder of magnitude smaller than λc in Tl, In and Rb; wetherefore do not find a stable antiferromagnetic (AFM)solution by LDA for any of the M2Mo6Se6 compounds.

However, AFM ordering is also favored by the Coulombrepulsion U , which is much larger than Hund’s cou-pling I. Constrained LDA calculations give U=5.1 eVand J=0.62 eV per Mo atom in Tl2Mo6Se6. Withthese values, we estimate a magnetic coupling constantλU=0.12 for Tl,In and 0.11 for Rb, which implies that allM2Mo6Se6 are on the verge of a SDW transition due tostrong electronic correlations.

For both charge (Peierls) and spin (AFM) densitywave transitions, the stronger tendency of Rb2Mo6Se6towards insulating behaviour compared with Tl2Mo6Se6

6

and In2Mo6Se6 is due to the smaller value of the criti-cal coupling constant λc. This is mainly caused by thereduced in-plane bandwidth of the helix band, which werecall is one order of magnitude smaller for M = Rbthan for M = In, Tl. Further details of our calcula-tions and the analytical model employed will be discussedelsewhere.28 We will now turn to our experimental data,which confirms our experimental calculations and pro-vides further insight towards the metal-insulator transi-tion.

III. EXPERIMENT

A. Sample Preparation and Experimental Details

Needle-like crystals of dimensions approximately4 mm×300 µm×100 µm and mass ≈ 800 µg were syn-thesised by different methods depending on the thermalstability of the compounds: Tl2Mo6Se6 and Rb2Mo6Se6were prepared in sealed molybdenum crucibles at 1700 Cand 1500 C respectively, whereas In2Mo6Se6 was pre-pared in an evacuated sealed silica tube at 1100 C. Theircrystalline structures were verified by the mono-crystaldiffraction method using a KAPPA CCD NONIUS andexhibit a slight cation deficiency with occupancy factors0.95, 0.94 and 0.93 for In, Tl and Rb based compounds re-spectively. Larger polycrystalline samples of Tl2Mo6Se6and In2Mo6Se6 of mass 10-75 mg were also produced bythe same methods.

The samples were initially characterized by AC Suscep-tibility, measured using a Quantum DesignTM PhysicalProperty Measurement System (QD PPMS). As shownin Fig. 5, Tl2Mo6Se6 and In2Mo6Se6 both exhibit su-perconducting transitions at Tc=4.2 K and 2.85 K, with∆Tc=0.7 K and 0.5 K respectively. Rb2Mo6Se6 displaysa constant weak diamagnetic signal down to the lowesttemperature measured (1.7K) with no sign of supercon-ductivity being observed. No significant increase in tran-sition width was observed for the polycrystals comparedto the single crystals, indicating a high sample quality.We stress that all members of the M2Mo6Se6 family re-main stable under atmospheric conditions and none ofour samples has exhibited any ageing effects.

AC resistivity was measured using the same QD PPMSwith a Helium-3 insert from 0.35-300 K. Four gold con-tacts of thickness ∼5 nm were sputtered onto single crys-tals of each compound and 50 µm gold wires attachedusing silver epoxy glue. This method yielded contact re-sistances ∼1 Ω. Short (1 s) pulses of a small AC current(0.02 mA, 470 Hz) were used to minimise any heatingeffects in the sample at low temperature.

Specific heat was initially also measured in the QDPPMS using a standard relaxation technique from 0.35-300 K. The largest homogeneous polycrystalline samplesavailable for each compound were mounted using a mea-sured quantity of WakefieldTM grease (whose contribu-tion to the heat capacity was later subtracted). How-

FIG. 5: AC susceptibility of Tl2Mo6Se6, In2Mo6Se6 andRb2Mo6Se6 from 1.8-300 K. Inset: zoom onto superconduct-ing transitions, with Tc=4.2 K and 2.85 K in Tl2Mo6Se6 andIn2Mo6Se6 respectively. As is the case for all graphs in thiswork, only a small fraction (typically 10-20 %) of the data-points measured are explicitly marked for clarity.

ever, due to the extremely small density of states at theFermi level the QD PPMS was unable to detect the su-perconducting transition in Tl2Mo6Se6 and In2Mo6Se6.High-sensitivity relaxations from 1.3-10 K were thereforecarried out on single crystals in our dedicated specificheat laboratory, enabling us to study the superconduct-ing transition with a magnetic field both perpendicularand parallel to the z axis.

B. Characteristics of the Superconducting State inTl2Mo6Se6 and In2Mo6Se6

As previously reported,5 resistive transitions into thesuperconducting state of Tl2Mo6Se6 exhibit an anoma-lous broadening under an applied magnetic field H, sim-ilar to that seen in the HTS. We have performed similarmeasurements on In2Mo6Se6 and show these in Fig. 6,together with the results in Tl2Mo6Se6 for comparison.It can immediately be seen that In2Mo6Se6 also displaysa broadening of the transition with increasing field, al-though the effect is less spectacular than in Tl2Mo6Se6.This is highlighted in Fig. 6(f), where we have plotted thenormalised transition width ∆Tc(H)/∆Tc(0) as a func-tion of normalised perpendicular magnetic field H/Hc2

for both Tl2Mo6Se6 and In2Mo6Se6. Both materials dis-play a linear behaviour in ∆Tc(H)/∆Tc(0) as the fieldincreases, with the gradient for Tl2Mo6Se6 ∼ 30% largerthan that of In2Mo6Se6.

The width of a superconducting transition is gov-erned by two parameters: the intrinsic homogeneity ofthe superconductor and the narrow thermal fluctuation-dominated critical region which surrounds any phase

7

FIG. 6: (a)-(e): Resistive transitions in Tl2Mo6Se6 andIn2Mo6Se6 with field both parallel and perpendicular to thez axis. Two samples of Tl2Mo6Se6 are shown: Sample (1)displaying a double transition indicating an inhomogeneousTl content and Sample (2) with a single broad transition. (f):Normalised transition widths ∆Tc(H)/∆Tc(0) as a functionof normalised magnetic field H/Hc2. For a transition in mag-netic field H, ∆Tc is defined as the temperature differencebetween resistivities of 5% and 95% of the saturated normal-state value at 7 K.

transition. Inhomogeneity contributions are field-independent, but for 3D fluctuations the critical region,whose width is given by the Ginzburg parameter G3D =(kbTc/

√2ξ2⊥ξ‖Hc(0)2)2 multiplied by Tc, is weakly field-

dependent, since G3D(H) ≈ G1/33D (H/Hc2)2/3. This

clearly cannot explain our data, since we observe ∆Tcto increase linearly as H increases. However, Mishonovet al.29 derived a quasi-1D Ginzburg parameter for a su-perconducting nanowire:

G1D =kB

8√π∆Cξ(0)S

(6)

where ∆C is the specific heat jump at Tc, S is thecross-sectional area of the nanowire and we only con-sider longitudinal fluctuations so ξ(0) ≡ ξ‖(0). ModellingM2Mo6Se6 as a weakly-coupled assembly of supercon-ducting filaments, each with the radius of a single Mo6Se6chain, we calculate critical region widths G1DTc = 1.5 Kand 2.0 K for Tl2Mo6Se6 and In2Mo6Se6 respectively.

Similar trends are also seen in the electronic contri-bution to the heat capacity (Fig. 7), with Tl2Mo6Se6and In2Mo6Se6 behaving very differently in a magnetic

FIG. 7: (a)-(c): Superconducting transitions in Tl2Mo6Se6and In2Mo6Se6 seen by specific heat with field both paralleland perpendicular to the z axis. (d),(e): BCS fits of the zero-field transition in Tl2Mo6Se6 and In2Mo6Se6.

8

field. The onset temperature Tons of the specific heatjump in In2Mo6Se6 is rapidly and uniformly displacedto lower temperature with increasing H, whereas regard-less of the applied field strength or orientation, Tons doesnot drop below ∼ 4K in Tl2Mo6Se6. It should be notedthat In2Mo6Se6 shows an abnormally large electroniccontribution to the specific heat γ in proportion to itsspecific heat jump, suggesting that only around 50% ofthe electrons at the Fermi level become superconducting.This scenario could be explained by a slight variation inthe sample stoichiometry leading to a CDW coexistentwith superconducting regions in the same crystal, hencealso supporting our observation that In2Mo6Se6 is moreanisotropic than Tl2Mo6Se6. Correcting for this anomalyin In2Mo6Se6, we estimate a Sommerfeld constant γ ≈0.13 mJgat−1K−2 for both Tl2Mo6Se6 and In2Mo6Se6,corresponding to a dressed density of states at the Fermilevel DEF

= 0.055 states eV−1 atom−1.We have attempted to fit the low-temperature spe-

cific heat in Tl2Mo6Se6 and In2Mo6Se6 using a standardBCS s-wave single-band α-model, as shown in Fig. 7(d).As discussed in the previous section, our band struc-ture calculations indicate that only a single band (the1D Mo d helix) crosses the Fermi level, thus eliminat-ing any possibility of multi-band superconductivity inM2Mo6Se6. Although the jump we measure at Tc is ther-mally broadened in each case, this yields an excellent fitfor In2Mo6Se6 with a gap value of 0.4 meV correspond-ing to 2∆0/kBTc = 3.5, just below the standard weak-coupling BCS value 3.52. An unusual hump on the backof the peak in Tl2Mo6Se6 renders the fit more difficultin this compound, but by using entropy conservation itis still possible to estimate a gap ∆0 ≥ 1.1 meV and2∆0/kBTc ≥ 5, placing it in the extreme strong-couplingregime. Although the quality of our fit in In2Mo6Se6does not appear to favour the presence of low-lying ex-citations, we were not able to accurately measure to suf-ficiently low temperatures in Tl2Mo6Se6 to conclusivelyrule out the existence of d-wave superconductivity in thismaterial. However, it is clear that Tl2Mo6Se6 has a sig-nificantly less conventional superconducting ground statethan In2Mo6Se6, an astonishing difference consideringthe close similarity between the compounds.

Completing the analysis of our data using anisotropicGinzburg-Landau theory,30 we summarise the supercon-ducting parameters of both Tl2Mo6Se6 and In2Mo6Se6in Table III. The large calculated values for κ highlightboth the extreme type II nature of these superconductors

and their enormous anisotropy. H‖c2 and ξ‖ should be re-

garded as minima, due to the high sensitivity of thesematerials to the field orientation.6 We estimate a samplealignment better than ± 2 with the field; however non-parallel crystalline intergrowths may exist within a single

needle-like sample which would reduce our measuredH‖c2.

The measured H‖c2 = 4.35 T in In2Mo6Se6 approaches the

Clogston limit HP = 4.9 T and it would be instructiveto re-measure the resistivity of a small single crystal in ahigh-accuracy goniometer in order to verify the possibil-

TABLE III: Measured and derived anisotropic superconduct-ing parameters in Tl2Mo6Se6 and In2Mo6Se6

Tl2Mo6Se6 In2Mo6Se6‖ ⊥ ‖ ⊥

MeasuredTc 4.2 K 2.85 KHc2(0) 5.9 T 0.47 T 4.35 T 0.25 THc(0) 0.0207 T 0.0119 T

Derivedε 12.6 17.2ξ(0) 940 A 75 A 1500 A 87 Aλ(0) 0.12 µm 1.5 µm 0.13 µm 2.2 µmκ 202 1.3 260 0.87G3D 3.3 10−6 3.0 10−6

G1D 0.36 0.69γn 0.13 mJ K−2 gat−1 0.13 mJ K−2 gat−1

BCS gap ∆0 ≥ 0.9 meV 0.4 meV2∆0/kBTc ≥ 5 3.5

HP = ∆0/√

2µB 11 T 4.9T

ity of Pauli-limited superconductivity occurring in thiscompound.

C. Normal-State Specific Heat and PhononDensities of States

Two features are immediately apparent in a plot ofCtot/T vs. T : a strong peak at T∼ 80 K and a shoulderat T∼ 20 K, indicating two dominant phonon energies.

FIG. 8: Total specific heat divided by temperature C/T forTl2Mo6Se6 (circles), In2Mo6Se6 (squares) and Rb2Mo6Se6(diamonds). Note that the increased noise level above 100 Kin Rb2Mo6Se6 can largely be attributed to a low sample mass.Inset: Close-up of C(T ) in Rb2Mo6Se6, showing a small peakbetween 170 K and 185 K.

The high-temperature specific data measured on largerpolycrystalline samples are sufficiently noise-free to per-

9

mit an inversion of the phononic contribution Cph(T ) tothe total heat capacity, hence obtaining the PDoS F (ω).We stress that this method does not provide a detailedPDoS map of the type revealed by neutron scattering, butrather produces a smoothed phonon distribution func-

tion F (ω). The specific heat and low-temperature fea-

tures of the PDoS are accurately reproduced by F (ω),but it does not give a high-precision representation ofthe PDoS at high temperature. We model F (ω) as a se-ries of logarithmically-spaced Einstein modes with fixedenergies and adjustable weights:

FIG. 9: (a) Lattice specific heat divided by temperatureC/T −γ for In2Mo6Se6 (triangles) fitted (solid line) by an as-sembly of Einstein modes (dashed lines). (b) Lattice specificheat normalised by temperature cubed C−γT/T 3 (triangles),highlighting the quality of our low temperature fit (solid anddashed lines) and the dominant contribution from the opticalmode generated by the In ion.

F (ω) =∑k

Fkδ(ω − ωk) (7)

Using this representation, the lattice specific heat is

then given by

C(T ) = 3R∑k

Fkx2ke

xk

(exk − 1)2(8)

where xk = ωk/T and ωk+1/ωk = 1.75 to limit thenumber of modes and provide a stable solution. A least-squares fit of the measured specific heat for each com-pound was performed and the decomposition into Ein-stein modes shown for In2Mo6Se6 as an example in Fig. 9.The fitting technique for Tl2Mo6Se6 and Rb2Mo6Se6 isidentical and the results of a similarly high quality, accu-rately reproducing our experimental data over the entiretemperature range.

FIG. 10: Measured phonon density of states for Tl2Mo6Se6,In2Mo6Se6 and Rb2Mo6Se6 (histograms) plotted simultane-ously with PDoS data from neutron scattering (crosses).

The fitted PDoS are shown for each compound inFig. 10 and the results obtained compare very favourablywith previous neutron scattering data.18 All three com-pounds clearly display the two key features already iden-tified in the C/T plots: a strong narrow peak below ∼10 meV and a broader distribution of phonons from ∼10-30 meV. The low-energy peak corresponds to an op-tical mode - the “guest” ion phonon - which is mediated

10

by the M ions vibrating in the tubes formed between theMo6Se6 chains. The broader maximum at higher energyis mainly due to acoustic intra-chain phonons, includingthe two Peierls modes discussed earlier.

Comparing the PDoS of each compound measured, thetwo superconductors are very similar with a slight spec-tral weight shift to higher energy in the optical cationmode for In2Mo6Se6. This can be explained by consider-ing the smaller atomic mass of In compared to Tl. How-ever, the intensity of the optical mode in Rb2Mo6Se6 issignificantly reduced and its frequency has been shiftedto ∼8 meV, compared with ∼4-6 meV in Tl2Mo6Se6and In2Mo6Se6. Neutron scattering data also showed ex-tensive hybridization of the cation mode with the chainmodes in Rb2Mo6Se6, thus inducing a deformation in thePDoS from 10-30 meV which can also be seen in our data.

D. Normal-State Resistivity and Electron-PhononCoupling

The resistivity curves from 0.35-300 K for each com-pound are shown in Fig. 11. Tl2Mo6Se6 and In2Mo6Se6both show linear metallic behaviour in the normal state.There is no evidence for any negative curvature result-ing from strong correlation effects or the existence of aLuttinger Liquid ground state at high temperature.31 Asaturation in the resistivity is observed for T < 15 K inTl2Mo6Se6 and In2Mo6Se6, with residual resistivity ra-tios of 10.1 and 8.8 respectively. Conversely, Rb2Mo6Se6undergoes a broad metal-insulator transition with a mini-mum at Tc = 170 K and an activation energy EA = 173K.

FIG. 11: Resistivity in M2Mo6Se6 normalised to 300K. Inset:low-temperature resistivity in Tl2Mo6Se6 and In2Mo6Se6 nor-malised to T 3 after residual subtraction, highlighting the largecontribution from low-energy phonons in Tl2Mo6Se6.

Using the same phonon frequency bins as those used tocalculate the PDoS, we may now evaluate the electron-phonon coupling from the normal-state resistivity data

FIG. 12: Resistivity (with residual subtracted) normalised byT and fitted with the same Einstein mode energies used forthe PDoS determination from our specific heat data.

for Tl2Mo6Se6 and In2Mo6Se6 (it is not possible to per-form this analysis on Rb2Mo6Se6, due to the metal-insulator transition). This procedure has been success-fully used to obtain the electron-phonon coupling in thesuperconducting borides ZrB12,21 YB6

22 and LuB1232

as well as the clathrate superconductors Ba8Si46 andBa24Si100;33 a more detailed account of the method canbe found in the references. Our departure point is thegeneralised Bloch-Gruneisen formula:34

ρBG(T ) = ρ(0) +4πm∗

ne2

∫ ωmax

0

α2trFω

xex

(ex − 1)2dω(9)

where x ≡ ω/T and α2trFω is the electron-phonon

“transport coupling function” which can be decomposedinto Einstein modes to give

α2trFω =

1

2

∑k

λtr,kωkδ(ω − ωk) (10)

Substituting this back into equation 9 yields the dis-crete version of the Bloch-Gruneisen equation:

11

FIG. 13: Phonon density of states (histogram) and normalisedelectron-phonon transport coupling function α2

trFω (line) forTl2Mo6Se6 and In2Mo6Se6.

ρBG(T ) = ρ(0) +2π

ε0Ω2p

∑k

λtr,kωkxke

xk

(exk − 1)2(11)

where the mode weighting parameters are the di-mensionless constants λtr,k. The residual resistiv-ity ρ(0) is determined separately from the raw data,equalling 39.5 µΩ cm for Tl2Mo6Se6 and 37.2 µΩ cmfor In2Mo6Se6.

Our fits are shown in Fig. 12 and display a clear dif-ference between Tl2Mo6Se6 and In2Mo6Se6. The initialslope of (R − R0)/T is much steeper in Tl2Mo6Se6, dueto a large contribution from a mode with energy 59 K. Incontrast, the lowest energy mode contributing to the re-sistivity in In2Mo6Se6 is centred at 103 K, with the firstsignificant contribution only arriving at 180 K. In theabsence of any data in the literature for the carrier den-sity n, we decompose the unscreened plasma frequencyΩ2

p = ne2/ε0m∗ and express our fitted values for α2

trFω

normalised by ne2/m∗. These are displayed in Fig. 13,superimposed on the PDoS.

The principal electron-phonon coupling for each com-pound occurs in the 10-18 meV frequency window. How-ever, Tl2Mo6Se6 also exhibits a major coupling in the3.5-6 meV region, in direct contrast with In2Mo6Se6,which shows no coupling at all below 6 meV. α2

trFω isintimately related to α2Fω, the electron-phonon couplingfunction governing superconductivity,35 implying an ad-ditional electronic coupling to the low-energy guest ionphonon in Tl2Mo6Se6.

IV. DISCUSSION

A. Superconducting Transitions

Tl2Mo6Se6 and In2Mo6Se6 both display an anoma-lous broadening of their resistive superconducting transi-tions, whose amplitude and variation under applied fieldis not compatible with conventional 3D thermal fluctua-tion models. This suggests that the quasi-1D nature ofthese compounds has a significant influence on the size ofthe critical region around Tc. In 3D systems, the broad-ening of a superconducting transition under applied mag-netic field is due to a finite size effect:36 the vortex-vortexseparation limits the divergence of the correlation lengthat Tc, hence reducing the coherence volume and increas-ing the importance of thermal fluctuations. In contrast,a perfect 1D system cannot undergo a phase transitiondue to insufficient degrees of freedom. Tl2Mo6Se6 andIn2Mo6Se6 lie in the crossover regime between these twoextremes.

Developing a theoretical model for this transition re-gion is a difficult task; however, we may consider thePeierls transition as an analogous crossover from a quasi-1D system to a quasi-3D ordered state. Theory predictsa suppression of Tc by a factor of up to 4, together witha light smearing of the transition37 and certain Peierlssystems indeed exhibit significantly broadened “jumps”in their resistivity as a result of quasi-1D fluctuations.38

It therefore seems reasonable to attribute the broaden-ing seen in Tl2Mo6Se6 and In2Mo6Se6 to the extremelow-dimensional nature of the compound.

Explaining why the effect is so much more notice-able in Tl2Mo6Se6 than In2Mo6Se6, particularly in thespecific heat jump, is rather harder especially giventhat In2Mo6Se6 is more anisotropic. The calculated 3DGinzburg number G3D for Tl2Mo6Se6 is only 10% largerthan that of In2Mo6Se6 and, in any case, G3DTc is sev-eral orders of magnitude too small to explain the ob-served broadening. Calculating the 1D Ginzburg param-eters G1D from the Mishonov model yields more realistictransition width amplitudes, although the fact that themeasured transition width for Tl2Mo6Se6 is larger thanthat of In2Mo6Se6 implies that this sample was less in-trinsically homogeneous.

The evolution of G1D with applied magnetic field hasnot yet been calculated for a superconducting nanowireand we were hence unable to track the broadening ofthe resistive transition using a low-dimensional model.However, we expect that a field-induced finite size effectsimilar to that seen in 3D systems should control thetransition widths. The crucial factor here is the differ-ence in coherence volumes ξ2⊥ξ‖ between the two super-

conductors: 5290 nm3 for Tl2Mo6Se6 and 11400 nm3 forIn2Mo6Se6. It is well known that low-dimensional fluc-tuations play an increasingly important role upon thereduction of coherence length in a material; the smallercoherence volume in Tl2Mo6Se6 must therefore outweighits lower anisotropy relative to In2Mo6Se6.

12

It is clear that our understanding of low-dimensionalfluctuations at a superconducting transition wouldgreatly benefit from a detailed theoretical analysis. Inparticular, the reproducible deformation of the specificheat jump in Tl2Mo6Se6 (which presumably results froma displacement of states from above Tc to the back ofthe jump) is a remarkable phenomenon and merits fur-ther attention. The shape of the jump is reminiscentof that seen in superconducting carbon nanotube ma-trices,39 suggesting that such behaviour may be intrin-sic to weakly-coupled superconducting filamentary net-works. A Berezinski-Kosterlitz-Thouless transition isalso thought to occur in such materials.40 Within thismodel, the superconducting transition is broadened dueto the appearance of a phase-incoherent intermediatestate (consisting of Josephson-coupled quasi-1D super-conducting fibres) separating the globally-coherent su-perconducting ground state at low temperature fromthe metallic normal state above Tc. Tl2Mo6Se6 andIn2Mo6Se6 would be prime candidates to undergo a sim-ilar transition.

We note that despite the strong evidence for quasi-1D fluctuations around the superconducting transitionsin Tl2Mo6Se6 and In2Mo6Se6 (as well as the broad re-sistivity minimum corresponding to the metal-insulatortransition in Rb2Mo6Se6), there is no indication of ahigh-temperature Luttinger Liquid ground state in bulkM2Mo6Se6. Transport experiments on Mo6Se6 nanowireshave revealed Luttinger behaviour in their conductance,which vanishes as the wire diameter increases above sev-eral tens of nanometers.41 However, since we are mea-suring M2Mo6Se6 crystals with diameters of the order ofseveral hundred microns (and hence an increase in thenumber of conduction channels by a factor of 108), theabsence of Luttinger effects is to be expected.

B. Electron-phonon coupling

Our BCS s-wave fits of the specific heat below Tc dis-play conventional weak coupling (2∆0/kBTc = 3.4) forIn2Mo6Se6 and extremely strong coupling (2∆0/kBTc ≥5) for Tl2Mo6Se6. In fact, Tl2Mo6Se6 may wellhave usurped the throne of the β-pyrochlore KOs2O6

(2∆0/kBTc ≥ 5) as the strongest-coupling phonon-mediated superconductor currently known. It should benoted that abnormally strong coupling (ranging up to2∆0/kBTc ∼ 25) is a characteristic of several quasi-1DCDW systems such as NbSe3 and (TaSe4)2I,42,43 due tothe transition temperature being suppressed below itsmean-field value. However, given that In2Mo6Se6 is moreanisotropic than Tl2Mo6Se6, we do not believe that thestrong coupling in Tl2Mo6Se6 originates from its low di-mensionality.

Deconvolving the normal-state resistivity shows thatthe predominant common electron-phonon coupling forTl2Mo6Se6 and In2Mo6Se6 lies in the 10-18 meV range,implying that superconductivity is mediated by the intra-

chain modes which (according to neutron diffraction ex-periments18) range from 12-40 meV and peak strongly at17 meV. This interpretation is supported by a tentativereport of superconductivity under pressure in Mo6Se6.4

The additional coupling to the low-energy optical modein Tl2Mo6Se6 moves this superconductor into the ex-treme strong-coupling regime, increases Tc by nearly 2 Kand reduces the coherence volume ξ2⊥ξ‖, rendering the su-perconducting transition more susceptible to broadeningthrough quasi-1D fluctuations.

Our observation immediately begs the question whyIn2Mo6Se6 does not enjoy a similar coupling to its op-tical In+ mode. There are two reasons for this: firstly,consider the variation of the hexagonal lattice parametera and the Pauling radii Rp of the Tl+, In+ and Rb+

monovalent cations. Values for a measured by X-raydiffraction1,44 in Tl2Mo6Se6, In2Mo6Se6 and Rb2Mo6Se6are given in Table IV, together with standard Rp values

from the literature. Each cation is located at a/√

3 from3 equidistant Mo6Se6 chains and, since the chain radiusis invariant with respect to the cation, the ratio

√3Rp/a

gives a good measure of the freedom of the respectivecations to vibrate in their inter-chain tunnels.

TABLE IV: Lattice parameters a and cation radii Rp inM2Mo6Se6

a (A) Rp (A)√

3Rp/aTl2Mo6Se6 8.94 1.15 0.223In2Mo6Se6 8.85 1.04 0.204Rb2Mo6Se6 9.26 1.48 0.277

It can clearly be seen that the In+ ion is less geomet-rically constrained than Tl+ and that Rb+ is at consid-erably less liberty to vibrate than either of its “super-conducting” counterparts. This is evident in the neu-tron scattering data: as pointed out by Brusetti et al.,Rb2Mo6Se6 displays significant hybridisation of the low-energy Einstein phonon, with the higher-energy inter-nal chain modes corresponding to a ∼40% increase inthe M ion force constants. Upon closer examination,the neutron-imaged PDoS of In2Mo6Se6 has a slightlydeeper trough at ∼11 meV than Tl2Mo6Se6, implyingmarginally less phonon hybridisation, which is consis-tent with our estimate above. We therefore believe thatthe interchain tunnel diameter in In2Mo6Se6 is simplytoo large relative to the In+ ion to allow its low-energyphonon to effectively couple to the Mo d electrons at theFermi level in the chains.

Secondly, the intrinsic electron-phonon couplingstrength λ is proportional to 1/ω2, where ω is the char-acteristic phonon frequency.34 Due to its smaller mass,the In+ mode is shifted to higher energy as can be seenboth in our PDoS histograms and the neutron data fromBrusetti in Fig. 10. Using the Tl+ = 5.2 meV andIn+ = 6.3 meV mode energies from Brusetti et al.18, wecalculate ω2

In/ω2Tl = 1.47; i.e. the coupling to the Tl+

mode should be nearly 50% stronger than that to the

13

In+ mode. To make a very crude comparison, we sumour measured α2Fω from Fig. 13 in the relevant energyrange 5.1 - 8.9 meV, obtaining Σ α2Fω(Tl)/Σ α2Fω(In)= 2.04. This suggests that the frequency-dependent vari-ation in coupling strength and the geometric constraintson the guest ion mode have a roughly equal importancein determining the coupling in M2Mo6Se6.

Naively, we might expect an enormous electron-phononcoupling and ultra-strongly-coupled superconductivity inRb2Mo6Se6 due to its narrow effective tunnel diameter.However, our electronic structure calculations indicatethat it is not the geometric constraints on the M ionin M2Mo6Se6 which determine its anisotropy, but ratherthe degree of warping in its Fermi sheets. This is con-trolled by the in-plane bandwidth w, as detailed in Ta-ble II. The electropositivity of the M ion - its willingnessto donate electrons - is a useful quantity for character-ising the behaviour of M2Mo6Se6, since it is inverselyproportional to w. Group IA metals are much more elec-tropositive than Group III and hence Rb2Mo6Se6 fallsvictim to a high temperature insulating instability ratherthan becoming superconducting at low temperature. (Itshould nevertheless be noted that high-pressure measure-ments have succeeded in partially suppressing the metal-insulator transition in Rb2Mo6Se6 and simultaneously in-ducing superconductivity with a maximum Tc ∼ 4 K.4)In a similar fashion, In is more electropositive than Tl,thus explaining the increased anisotropy seen in the su-perconducting state of In2Mo6Se6 compared to that ofTl2Mo6Se6.

C. Metal-Insulator Transitions in M2Mo6Se6

Our LDA calculations have indicated two possi-ble mechanisms for the metal-insulator transition inM2Mo6Se6: a Peierls transition resulting in the forma-tion of a dynamical CDW, or a SDW driven by strongCoulomb repulsion. Let us initially consider the SDWscenario.

SDWs and their associated low-temperature antiferro-magnetic order are typically imaged using neutron scat-tering techniques. Unfortunately, no such data exist inthe literature for any of the M2Mo6Se6 family. How-ever, a SDW may also be detected by the temperaturedependence of its magnetic susceptibility χ(T ). In aSDW, χ(T ) generally exhibits paramagnetic behaviourabove the transition, but then displays a characteristicmaximum at the critical temperature TMI before falling,hence signalling the onset of antiferromagnetism. AnySDW transition should therefore be clearly visible in ourac susceptibility data (Fig. 5). Examining this closely, noevidence can be seen for any departure from temperature-invariant weakly-diamagnetic behaviour above the noisethreshold of our susceptometer (10−8 emu). In partic-ular, the ac susceptibility of Rb2Mo6Se6 does not dis-play any anomaly as it passes through TMI . We there-fore find no experimental evidence for SDW formation in

M2Mo6Se6.

Note that our calculated magnetic coupling constantsλU are greater than the critical couplings λc for allM2Mo6Se6, not just Rb2Mo6Se6. This implies that anySDW transition should be present in all three com-pounds. However, no TMI may be identified in ourresistivity data for Tl2Mo6Se6 and In2Mo6Se6. Fur-thermore, it is not realistic to suggest that quasi-1D fluctuations suppress SDW formation in Tl2Mo6Se6and In2Mo6Se6 but not Rb2Mo6Se6: from our elec-tronic structure calculations, Rb2Mo6Se6 is much moreanisotropic than Tl2Mo6Se6 and In2Mo6Se6 and shouldhence be more susceptible to such fluctuations. A farmore probable scenario is that low-dimensional fluctua-tions prevent the SDW transition from taking place in allthree compounds, with the metal-insulator transition inRb2Mo6Se6 driven by a separate mechanism. Our LDAcalculations for the electron-phonon coupling point to-wards this being a dynamical CDW and we will continueour discussion from this perspective.

The transition seen in the electrical resistivity ofRb2Mo6Se6 is very broad and continuous compared withthe distinct steps seen in other CDW materials such asNbSe3 or TaS2.38,45 Such “blurring” can be explainedby a combination of the influence of low-dimensionalphase fluctuations and the dynamical nature of an un-derlying CDW whose order parameter may vary in bothtime and space. Tarascon2 and Hor3 have also per-formed transport measurements on Rb2Mo6Se6, obtain-ing similar broad transitions with TMI ∼ 125 K, 104 Kand EA = 87 K, 145 K respectively, which are rathersmaller than our values TMI ∼ 170 K and EA =173 K. We attribute the differences between TMI to vari-ations in sample anisotropy, resulting from slight dif-ferences in the stoichiometries. It is well-known thatTl2Mo6Se6 is not a perfectly stoichiometric compound,12

with the highest TMI occurring in Tl1.95Mo6Se6. Sim-ilar non-stoichiometric behaviour occurs in the rest ofthe M2Mo6Se6 family, for example producing occasionalbatches of In2Mo6Se6 which are non-superconducting.9,44

No clear evidence has been seen for the metal-insulatortransition by any other experimental technique, althoughwe observe a small anomaly in the heat capacity ofRb2Mo6Se6 between 170 K and 185 K (see Fig. 8), justabove TMI . However, the height of the peak scarcelyexceeds the noise threshold of our data and higher res-olution measurements will be required to verify andquantify this feature. Given the limited evidence for athermodynamic phase transition at TMI , it is temptingto hold intra-chain defects responsible for the insulat-ing behaviour observed at low temperature, relying onthermally-activated interchain hopping to enable metallicconductivity at high temperature. However, this simplemodel cannot be applicable to M2Mo6Se6 since it is un-able to explain the broadband conductance noise indica-tive of sliding density wave motion.13 Furthermore, with-out a phase transition to deplete the DoS at the Fermilevel Rb2Mo6Se6 would become superconducting at low

14

temperature (unless the defect density is sufficiently highto enable weak localization). No trace of superconduc-tivity in Rb2Mo6Se6 has been seen by any experimentaltechnique at ambient pressure.

An examination of the behavioural trends in the re-maining (Group IA) members of the M2Mo6Se6 familylends further support to the CDW argument. Since theelectronic anisotropy is proportional to the electropos-itivity of the M atom, we would expect to see metal-insulator transition temperatures fall as we move upGroup IA (i.e. Cs→Na). Although no systematic studyof the family has been carried out by a single author,Tarascon et al.2 found that Cs2Mo6Se6 has a higher TMI

than Rb2Mo6Se6, which is in line with this hypothe-sis. Hor et al.3 have shown that hydrostatic pressure(leading to an enhanced interchain coupling and reducedanisotropy) suppresses TMI to lower temperatures, im-plying that TMI simply scales with the area of the Fermisurface available for nesting (as would be expected for aCDW). Nevertheless, a comprehensive study of all GroupIA members of the M2Mo6Se6 family will clearly be re-quired to validate our proposed dynamical CDW sce-nario. It is interesting to note that scanning tunnellingspectroscopy on small bundles of Mo6Se6 chains has re-vealed metallic behaviour down to 5 K;46 however thereduced electronic filling in Mo6Se6 results in a differentelectronic structure at the Fermi level with up to threebands occupied.47 A simple density wave model is henceunlikely to be applicable in this compound.

In addition to the obvious metal-insulator transitionin Rb2Mo6Se6, it is important to address the possibilityof superconductivity in Tl2Mo6Se6 arising from a hiddenCDW ground state, as suggested by the Hall effect resultsfrom Brusetti et al..12 Our results confirm that there isno discontinuity in the resistivity or jump in the spe-cific heat of either Tl2Mo6Se6 or In2Mo6Se6 to supportthis conjecture. Local stoichiometric heterogeneity inthe samples measured could produce a continuous seriesof local transitions, further blurred by quasi-1D fluctua-tions. However, this would produce a positive curvaturein ρ(T ) for T < 80 K, which is not seen. Alternatively, ifwe assume that the inter-chain coupling in Tl2Mo6Se6 issufficiently strong for such low-dimensional fluctuationsto be negligible, a theoretical model48 may be invokedwhich defines two separate transition temperatures: Tcuat which the CDW distortion occurs and Tcl at whichan energy gap opens over the entire Fermi surface and ametal-insulator transition occurs. Despite the presenceof a structural modulation, metallic behaviour persistsin the intermediate temperature range Tcl < T < Tcuwhose width is acutely dependent on the anisotropy. Us-ing this model and the Brusetti Hall coefficient data, wecould identify T ∼ 80 K as Tcu in Tl2Mo6Se6 and as-sume that the superconducting transition takes place atTc > Tcl. However, we cannot justify disregarding quasi-1D fluctuations (as required by the model) when they aremanifested so clearly in the deformation of the specificheat jump below Tc in Tl2Mo6Se6.

A more attractive explanation envisages a par-tial CDW gradually depleting the Fermi surface inTl2Mo6Se6 as the temperature decreases. This modelagrees perfectly with the observed crossover fromelectron-like to hole-like carriers, yet a corresponding in-crease in carrier mobility below 80 K is still requiredto explain the linear metallic resistivity. It is not clearwhat type of physical mechanism could be responsible forsuch a rise in mobility and hence the reported Hall co-efficient data in Tl2Mo6Se6 remains mysterious. In con-trast, the situation in Rb2Mo6Se6 is less complex: theobserved transition to insulating behaviour implies thata gap opens across the entire Fermi surface. However,with the present data set it is unfortunately impossible tojudge whether a similar “partial CDW” might be presentfor T > 170 K in this material.

Recent work49 on (TaSe4)2I has suggested the exis-tence of so-called unconventional CDWs (UCDWs) inquasi-1D systems, with a k-dependent gap which aver-ages to zero at the Fermi surface. This results in the for-mation of a pseudogap prior to a density wave transitionand is supported by photoemission data in K0.3MoO3 and(TaSe4)2I.50 Within the pseudogap phase, the UCDWcompetes with low-dimensional quantum fluctuations asdescribed by the Lee-Rice-Anderson model:37 these actto suppress the Peierls transition temperature TP wellbelow its mean-field value TMF , with the CDW en-ergy gap (the order parameter) fluctuating in time andspace between these temperatures. It seems likely thatM2Mo6Se6 displays a non-trivial combination of low-dimensional fluctuation effects and a dynamical CDWwith possible momentum dependence. Scanning tun-nelling spectroscopy (STS) or angle-resolved photoemis-sion spectroscopy (ARPES) would be ideal tools to verifythe existence of an UCDW, fluctuating order parameteror dynamical CDW since although a pseudogap shouldopen in the density of states, no static modulation of thestructure or charge density is expected.

V. CONCLUSIONS

We have calculated the electronic structures and stud-ied the resistivity and specific heat in both superconduct-ing and normal states for the quasi-1D M2Mo6Se6 familyof cluster condensates (M=Tl, In, Rb). These materialslie on the border between superconducting and insulatinginstabilities.

Superconductivity in Tl2Mo6Se6 and In2Mo6Se6 isprincipally mediated by an internal phonon from theMo6Se6 chains with an energy in the 12-18 meV range.It is hoped that theoretical calculations will soon iden-tify the precise energy and nature of this phonon.Tl2Mo6Se6 exhibits a further coupling to an optical modeat ∼5 meV, which we identify with interchain vibrationsof the Tl+ ion. This extra coupling is not present inIn2Mo6Se6 due to the higher In+ phonon energy andthe reduced geometrical constraints on the In+ ion be-

15

tween the Mo6Se6 chains. The low-temperature specificheat of In2Mo6Se6 is well-fitted by a standard single-gapisotropic s-wave BCS model with 2∆0/kBTc = 3.5±0.1.However, the additional interaction with the low-energymode in Tl2Mo6Se6 pushes it into the strong couplingregime with 2∆0/kBTc ≥ 5. The specific heat jump atTc also exhibits a significant deformation with a shift instates to low temperature, which we attribute to stronglow-dimensional fluctuations accentuated by the smallcoherence volume. STS or similar tunnelling experimentswould provide conclusive proof of the gap symmetry, aswell as displaying the coupling to the low-energy phonon.

Our LDA calculations show that in all members of theM2Mo6Se6 family, a single 1D helix band crosses theFermi level. Its in-plane dispersion w is reduced by afactor of ten in Rb2Mo6Se6 compared with Tl2Mo6Se6and In2Mo6Se6, while the out-of-plane dispersion is prac-tically unchanged. Using an analytical model, we haveshown that this reduction of w is sufficient to explain thetrend from metallic conductivity followed by strongly-

coupled superconductivity in Tl2Mo6Se6 to a high-temperature metal-insulator transition in Rb2Mo6Se6.This insulating ground state is a consequence of either adynamical Peierls (CDW) instability or a SDW transitiondriven by strong electronic correlations. The tempera-ture invariant magnetic susceptibility seen in all threecompounds favours a CDW interpretation. M2Mo6Se6are therefore an ideal target for future ARPES experi-ments, since a dynamical CDW should generate a pseu-dogap at the Fermi level.

VI. ACKNOWLEDGEMENTS

We thank A. Junod, M. Hoesch, Y. Fasano, C. Bern-hard and L. Forro for invaluable advice and discussions.This work was supported by the National Centre of Com-petence in Research MaNEP and the Swiss National Sci-ence Foundation.

∗ Electronic address: appetrovic@ntu.edu.sg1 M. Potel, R. Chevrel, M. Sergent, J. Armici, M. Decroux,

and Ø. Fischer, J. Solid State Chem. 35, 286 (1980).2 J. Tarascon, F. DiSalvo, and J. Waszczak, Solid State

Commun. 52, 227 (1984).3 P. Hor, W. Fan, L. Chou, R. Meng, C. Chu, J. Tarascon,

and M. Wu, Solid State Commun. 55, 231 (1985).4 P. Hor, R. Meng, C. Chu, J. Tarascon, and M. Wu, Physica

B+C 135, 245 (1985).5 A. Petrovic, Y. Fasano, R. Lortz, M. Decroux, M. Potel,

R. Chevrel, and Ø. Fischer, Physica C 2, 3 (2007).6 J. Armici, M. Decroux, Ø. Fischer, M. Potel, R. Chevrel,

and M. Sergent, Solid State Commun. 33, 607 (1980).7 H. Geserich, G. Scheiber, M. Durrler, M. Potel, M. Sergent,

and P. Monceau, Physica 143B, 234 (1986).8 I. Lee, P. Chaikin, and M. Naughton, Phys. Rev. B 65,

180502 (2002).9 R. Lepetit, P. Monceau, M. Potel, P. Gougeon, and M. Ser-

gent, Journal of Low Temperature Physics 56, 219 (1984).10 R. Brusetti, P. Monceau, M. Potel, P. Gougeon, and

M. Sergent, Solid State Commun. 66, 181 (1988).11 I. Lee, M. Naughton, G. Danner, and P. Chaikin, Phys.

Rev. Lett. 78, 3555 (1997).12 R. Brusetti, A. Briggs, O. Laborde, M. Potel, and

P. Gougeon, Phys. Rev. B 49, 8931 (1994).13 G. Tessema, Y. Tseng, M. Skove, E. Stillwell, R. Brusetti,

P. Monceau, M. Potel, and P. Gougeon, Phys. Rev. B 43,3434 (1991).

14 S. Huang, J. Mayerle, R. Greene, M. Wu, and C. Chu,Solid State Commun. 45, 749 (1983).

15 P. Kelly and O. Andersen, Superconductivity in d- and f-Band Metals (Academic Press: New York, 1982), pp. 134–140.

16 H. Nohl and O. Andersen, Superconductivity in d- and f-Band Metals (Academic Press: New York, 1982), pp. 161–165.

17 C. Dubois, A. Petrovic, G. Santi, C. Berthod, A. A.

Manuel, M. Decroux, Ø. Fischer, M. Potel, and R. Chevrel,Phys. Rev. B 75, 104501 (2007).

18 R. Brusetti, A. Dianoux, P. Gougeon, M. Potel, E. Bon-jour, and R. Calemczuk, Phys. Rev. B 41, 6315 (1990).

19 S. Bader, S. Sinha, B. Schweiss, and B. Renker, Super-conductivity in Ternary Compounds I (Springer-Verlag:Berlin, Heidelberg, New York, 1982), pp. 223–250.

20 E. Bonjour, R. Calemczuk, K. A.F., P. Gougeon, M. Potel,and M. Sergent, in Proceedings of the 2nd InternationalConference on Phonon Physics, Budapest, 1985, edited byJ. Kollar, N. Kroo, M. Menyhard, and T. Siklos (WorldScientific, Singapore, 1985), p. 570.

21 R. Lortz, Y. Wang, S. Abe, C. Meingast, Y. B. Paderno,V. Filippov, and A. Junod, Phys. Rev. B 72, 024547(2005).

22 R. Lortz, Y. Wang, U. Tutsch, S. Abe, C. Meingast,P. Popovich, W. Knafo, N. Shitsevalova, Y. B. Paderno,and A. Junod, Phys. Rev. B 73, 024512 (2006).

23 Z. Hiroi, S. Yonezawa, Y. Nagao, and J. Yamaura, Phys.Rev. B 76, 014523 (2007).

24 O. Andersen, Phys. Rev. B 12, 3060 (1975).25 P. Blaha, WIEN2k (K. Schwarz, TU Wien, Austria, 2001)

(ISBN 3-9501031-1-2, 2001).26 M. Potel, R. Chevrel, and M. Sergent, Acta Cryst. B36,

1545 (1980).27 B. Chew, J. Golden, B. Huggins, F. DiSalvo, and D. Zax,

Phys. Rev. B 50, 7966 (1994).28 L. Boeri and O. Andersen, in preparation.29 T. Mishonov, G. Pachov, I. Genchev, L. Atanasova, and

D. Damianov, Phys. Rev. B 68, 054525 (2003).30 J. Waldram, Superconductivity of metals and cuprates (In-

stitute of Physics Publishing, 1996).31 T. Giamarchi, Phys. Rev. B 44, 2905 (1991).32 J. Teyssier, R. Lortz, A. Petrovic, D. van der Marel,

V. Filippov, and N. Shitsevalova, Phys. Rev. B 78, 134504(2008).

33 R. Lortz, R. Viennois, A. Petrovic, Y. Wang, P. Toule-

16

monde, C. Meingast, M. Koza, H. Mutka, A. Bossak, andA. San Miguel, Phys. Rev. B 77, 224507 (2008).

34 G. Grimvall, The Electron-Phonon Interaction in Metals(North-Holland: Amsterdam, 1981), pp. 212,219.

35 P. Allen and R. Dynes, Phys. Rev. B 12, 905 (1975).36 R. Lortz, C. Meingast, A. Rykov, and S. Tajima, Phys.

Rev. Lett. 91, 207001 (2003).37 P. Lee, T. Rice, and P. Anderson, Phys. Rev. Lett. 31, 462

(1973).38 N. Ong and P. Monceau, Phys. Rev. B 16, 3443 (1977).39 R. Lortz, Q. Zhang, W. Shi, J. Ye, C. Qiu, Z. Wang, H. He,

P. Sheng, T. Qian, Z. Tang, et al., Proc. Natl. Acad. Sci.USA 106, 7299 (2009).

40 Z. Wang, W. Shi, T. Zhang, N. Wang, Z. Tang, X. Zhang,R. Lortz, P. Sheng, I. Sheikin, and A. Demuer, Phys. Rev.B 81, 174530 (2010).

41 L. Venkataraman, Y. Suk Hong, and P. Kim, Phys. Rev.Lett. 96, 076601 (2006).

42 A. Fournel, J. Sorbier, M. Konczykowski, and P. Monceau,

Phys. Rev. Lett. 57, 2199 (1986).43 D. Purdie, I. Collins, H. Berger, G. Margaritondo, and

B. Reihl, Phys. Rev. B 50, 12222 (1994).44 T. Mori, Y. Yokogawa, A. Kobayashi, Y. Sasaki, and

H. Kobayashi, Solid State Commun. 49, 249 (1984).45 J. Harper, T. Geballe, and F. DiSalvo, Phys. Rev. B 15,

2943 (1977).46 L. Venkataraman and C. Lieber, Phys. Rev. Lett. 83, 5334

(1999).47 F. Ribeiro, D. Roundy, and M. Cohen, Phys. Rev. B 65,

153401 (2002).48 C. Zhou and C. Gong, J. Phys. C: Solid State Phys. 21,

561 (1988).49 B. Dora, A. Vanyolos, and A. Virosztek, Phys. Rev. B 73,

125110 (2006).50 B. Dardel, D. Malterre, M. Grioni, P. Weibel, Y. Baer, and

F. Levy, Phys. Rev. Lett. 67, 3144 (1991).