We report measurements of the in-plane electrical resistivity $\rho$ and the
thermal conductivity $\kappa$ of the intercalated graphite superconductor
C$_6$Yb to temperatures as low as $T_c$/100. When a field is applied along the
c-axis, the residual electronic linear term $\kappa_0/T$ evolves in an
exponential manner for $H_{c1} < H < H_{c2}$. This activated behaviour
establishes the order parameter as unambiguously s-wave, and rules out the
possibility of multi-gap or unconventional superconductivity in this system.Comment: 4 pages, 4 figs, submitted to Phys. Rev. Let

Doiron-Leyraud, Nicolas

Ellerby, Mark

Saxena, S. S.

Sutherland, Mike

Taillefer, Louis

Weller, Thomas

English

arXiv

Mike Sutherland

Nicolas Doiron-Leyraud

Louis Taillefer

Thomas Weller

Mark Ellerby

S. S. Saxena

Crossref

Bulk Evidence for Single-Gap
                    
                      s

We report measurements of the in-plane electrical resistivity P and thermal conductivity K of the intercalated graphite superconductor C6Yb down to temperatures as low as Tc/100. When a field is applied along the c axis, the residual electronic linear term K0/T evolves in an exponential manner for Hc1<H<Hc2/2. This activated behavior is compelling evidence for an s-wave order parameter, and is a strong argument against the possible existence of multigap superconductivity

Sutherland, M.

Doiron-Leyraud, N.

Taillefer, L.

Weller, T.

Ellerby, M.

Saxema, S.S.

UCL Discovery

arXiv:cond-mat/0603664v1  [cond-mat.supr-con]  24 Mar 2006Bulk evidence for single-gap s-wave superconductivity in the intercalated graphitesuperconductor C6YbMike Sutherland1, Nicolas Doiron-Leyraud2, Louis Taillefer2,3,Thomas Weller4, Mark Ellerby4, S.S. Saxena11Cavendish Laboratory, University of Cambridge, J.J. Thomson Ave, Cambridge CB3 OHE, UK2Regroupement Qu´ eb´ ecois sur les Mat´ eriaux de Pointe,D´ epartement de physique, Universit´ e de Sherbrooke, Sherbrooke, Qu´ ebec, Canada3Canadian Institute for Advanced Research, Toronto, Ontarion, Canada4Department of Physics and Astronomy, University College London, Gower Street, London, UKWe report measurements of the in-plane electrical resistivity ρ and the thermal conductivity κ ofthe intercalated graphite superconductor C6Yb to temperatures as low as Tc/100. When a ﬁeld isapplied along the c-axis, the residual electronic linear term κ0/T evolves in an exponential mannerfor Hc1 < H < Hc2. This activated behaviour establishes the order parameter as unambiguouslys-wave, and rules out the possibility of multi-gap or unconventional superconductivity in this system.PACS numbers: 74.70.Wz, 74.25.Fy, 74.25.OpCarbon is a remarkably versatile element – in its pureform it may exist as an electronic insulator, semiconduc-tor or semimetal depending on its bonding arrangement.When dopant atoms are introduced, superconductivitymay be added to this list, observed in graphite [1, 2],fullerenes [3] and even diamond [4]. Superconductivity indoped carbon was ﬁrst discovered in the graphite interca-late compounds (GICs), materials composed of sheets ofcarbon separated by layers of intercalant atoms. The ﬁrstof these compounds were intercalated with alkali atoms,and had modest transition temperatures of 0.13-0.5 K[1]. The recent discovery of T ′cs two orders of magni-tude higher than this in C6Yb [5] and C6Ca [5, 6] hashowever refocused attention on this intriguing family ofcompounds.The eﬀects of the intercalant atoms in the GICs aretwo-fold: they dramatically change the electronic prop-erties of the host graphite lattice by both increasing theseparation of the carbon sheets, as well as contributingcharge carriers. This causes the two-dimensional graphiteπ∗ bands to dip below the Fermi level. The graphiteinterlayer band, previously unoccupied, also crosses thenew Fermi level, contributing three dimensional, free-electron like states located between the carbon sheets.This new interlayer band hybridises strongly with the π∗bands, and its occupation appears to be linked with theoccurrence of superconductivity in the GICs [7].There are still several fundamental questions remain-ing about superconductivity in the GICs, especially inC6Yb and C6Ca, where little experimental data exists.The pairing mechanism is unresolved, with speculationranging from a conventional route involving the inter-calant phonons [8, 9, 10] to superconductivity via acous-tic plasmons [7].Early theoretical studies motivated by the alkali-metalGICs [11, 12] emphasized a two-gap model for the su-perconducting state. In this picture, superconductiv-ity arises from coupling between intercalant s electronsand the graphite π electrons bands, with gaps of diﬀer-ent magnitudes existing on diﬀerent sheets of the Fermisurface. Such a scenario is plausible, as there are no-table similarities between the GICs and MgB2 [7, 13], aknown multi-gap superconductor. Indeed, some aspectsof graphite intercalate superconductivity can be under-stood by this two-gap phenomenology, however there islittle direct evidence to support this picture.The experimental starting point for addressing theseissues is to establish the superconducting order param-eter. In C6Yb and C6Ca, this task is complicated bythe diﬃculties in materials preparation arising from thestandard vapour transport process used for intercalation.This process typically yields samples which have a shellof fully intercalated material surrounding a core of un-intercalated graphite. In addition, both compounds areextremely sensitive to air, and their surfaces rapidly de-teriorate if left exposed. A recent study of penetrationdepth [14] on C6Ca suggested that the superconductiv-ity was s-wave, however this technique is extremely sur-face sensitive, and the results were dependent on surfacetreatment. With these considerations in mind, we turnto measurements of bulk thermodynamic properties toprobe the superconducting state.The technique of thermal conductivity is ideally suitedto the study of these materials. It is sensitive only todelocalized states, and in highly conductive systems suchas C6Yb the majority of the heat transport at low tem-peratures is provided by electrons, allowing us to easilyseparate out electronic and phononic contributions to theheat current. Most importantly, thermal conductivity isa bulk probe, only marginally aﬀected by small concen-trations of impurity phases.In this Letter we report measurements of low temper-ature thermal conductivity (κ) in C6Yb, which we use toestablish the nature of the superconducting order param-eter. The behaviour of κ as the superconducting stateis suppressed with a magnetic ﬁeld shows an activateddependence, clear evidence of s-wave superconductivitywith a single gap energy scale.Thermal transport was measured down to 60 mK in adilution refrigerator using a one heater, two thermometer2FIG. 1: In plane resistivity for C6Yb with H   c. For H= 0 T the superconducting transition is sharp with Tc =5.4 ± 0.4 K but broadens with applied ﬁeld. For H = 1 Tsuperconductivity is entirely suppressed, revealing a metallicnormal state with a residual resistivity ρ0 = 5.8 µΩcm.steady state technique. Magnetic ﬁelds from 0 to 1 Twere applied parallel to the c-axis and perpendicular tothe in-plane heat current. For measurements of κ(T) atconstant ﬁeld, the sample was cooled in ﬁeld from T >Tc to maximize homogeneity of the vortex lattice.Our samples were prepared by intercalating very purehighly oriented pyrolytic graphite (HOPG) using thevapour transport process described elsewhere [5]. In or-der to obtain samples consisting mainly of intercalatedmaterial, we ﬁrst cleaved the graphite along the ab-plane,and then took thin bars from the sides. The resultingsamples were rectangular platelets of dimensions approx-imately 1 mm × 0.5 mm × 100  m. Good quality electri-cal contacts were made using Dupont silver paint applieddirectly on the surface after cleaving, with all handlingand mounting done in a glove box under ﬂowing He inorder to preserve the quality of the samples.Figure 1 shows the in-plane resistivity of C6Yb in bothzero ﬁeld (inset) and as a function of ﬁeld applied alongthe c-axis (main panel). The residual resistivity ρ0, isobserved to be 4.5  Ωcm by extrapolating the zero ﬁeldcurve, and slightly larger than this using the 1 T curve.The magnetoresistance is comparatively weak in C6Yb[15], only 30 % by 1 T at 4 K, compared to a factorof 100 increase by only 0.2 T in pure graphite at T =5 K [16]. From the magnitude of ρ0 we may estimate theelectronic mean free path ℓe, assuming kF ∼ 0.5 ˚ A−1 [11]we get ℓe ≃ 1000 ˚ A at low temperatures in zero ﬁeld.Figure 2 shows the dependence of the thermal conduc-tivity on ﬁeld for H   c-axis. By plotting κ/T vs. TFIG. 2: Low temperature κ of C6Yb as a function of appliedﬁeld, with J  ab and H   c. The black line is the normalstate Wiedemann-Franz law expectation estimated from theresistivity in H = 1 T.we see that κ/T is almost constant, reﬂecting a dom-inant electronic contribution. Phonons are responsiblefor the small slope, which has negligible impact of theﬁeld dependence of the residual linear term, the subjectof this paper. The rise in κ with H is gradual at ﬁrst,then rapidly approaches the normal state value as Hc2is reached. By 1 T the sample is fully in the normalstate, and a comparison to the electrical resistivity viathe Wiedemann-Franz law is shown by the dashed line.In the normal state we see that this law is obeyed towithin 5 %, as expected for a metal in the elastic scat-tering regime.Extrapolating the zero ﬁeld data to T = 0 inFig.2 yields a small residual linear term κ0/T ≃0.3 mWK−2cm−1. It is tempting to attribute this asarising from nodal quasiparticles in the superconduct-ing state [17], as observed for example in d-wave super-conductors such as the high-Tc cuprates, where κ0/T= 1.41 mWK−2cm−1 for overdoped Tl-2201 with Tc =15 K [18]. An estimate of the magnitude of the linearterm however makes this scenario unlikely. In a nodalsuperconductor the size of κ0/T is determined by ratioof the quasiparticle velocities parallel (v2) and perpen-dicular (vF) to the Fermi surface near the nodes [17].In a two dimensional d-wave superconductor with a gapmaximum ∆0 and a density of n planes per unit cell ofheight c we may writeκ0T=k2B6ndkF￿vF∆0￿(1)assuming vF ≫ v2 [19]. Using an average vF ≃ 3.4 ×107 cm/s [12], kF ∼ 0.5 ˚ A−1 [11] and ∆0 = 2.14kBTc= 1.2 meV we expect κ0/T = 6.1 mWK−2cm−1 for thissystem. This is over an order of magnitude larger thanwhat we measure, and on this basis we rule out an uncon-3FIG. 3: Thermal conductivity of C6Yb at T = 300mK as afunction of applied ﬁeld. Hc1 and Hc2 are determined as thepoints where the conductivity deviates from a nearly ﬂat Hdependence at low and high ﬁelds respectively. The dottedline is a ﬁt to the expected behaviour for an s-wave supercon-ductor between Hc1 < H < Hc2/2 [21].ventional order parameter in C6Yb. We instead interpretthe ﬁnite κ0/T as arising from inclusions of pure graphitewhere full intercalation was not successful. The extrinsicorigin of the linear term is supported by the fact that ina second sample with a similar residual resistivity (ρ0 =4.4  Ωcm), κ0/T was seen to be half as large.Since thermal conductivity probes the bulk of a sam-ple, we may use κ as a robust check of the volume fractionof superconducting material. Since an s-wave supercon-ductor has a fully gapped Fermi surface in the supercon-ducting state, we would expect no contribution to κ0/Tfrom the intercalated material. The non-intercalatedgraphite has a residual resistivity of ∼ 6  Ωcm [20], andso conducts heat and charge with the same ability as bulkC6Yb in the normal state at low temperatures. Thus, theratio of κ0/T in the superconducting state to that in theﬁeld induced normal state should, to ﬁrst approxima-tion, yield the volume fraction of non-superconductinggraphite inclusions.Performing this simple analysis yields(κ0,H=0/T)/(κ0,H=1T/T) ≃ 7 %, a relatively smallfraction of the sample. These arguments demonstratethat good quality, essentially bulk samples of C6Ybcan be obtained by carefully selecting material fromthe edges of an intercalated platelet prepared using thevapour transport process.We now turn to analysing the ﬁeld dependence of κ.Fig. 3 shows κ/T(H) with T held constant at 300 mK.Given the small phonon contribution evident from theslope of the data inFig. 2, the conductivity in the presentﬁgure is largely due to electrons. Starting from H = 0we see that κ/T is close to zero, and essentially ﬂat up toH ≃ 0.025 T with a small conductivity arising from thenon-superconducting graphite regions. For higher ﬁelds asudden increase in κ/T is observed, which we interpret asthe onset of the vortex regime at H > Hc1. This agreesreasonably well with estimates of Hc1 = 0.04 T at lowtemperatures from magnetization measurements [5].As the ﬁeld is further increased, the conductivityevolves in an exponential manner, precisely what is ex-pected for transport in the mixed state of an s-wavesuperconductor. As vortices ﬁrst enter the sample atH > Hc1 the only quasiparticle states at T ≪ Tc arethose associated within the vortex cores [22]. When thevortices are far apart, these states remain localized, andare unable to contribute to heat transport. Increasingthe ﬁeld decreases the intervortex spacing d ∼pΦ0/Band the states begin to overlap, forming dispersive bandswhich yield a conductivity that grows exponentially withthe ratio of the vortex spacing to the coherence lengthd/ξ, κ ∝√Hexp(-αpHc2/H) where α is a constant.This dependence is readily observed in simple s-wave su-perconductors such as Nb [21] for Hc1 < H < Hc2/2,and is much diﬀerent than that in nodal superconduc-tors, where the conductivity is observed to increase as pH/Hc2 [18]. A ﬁt of our data to the simple s-waveform is shown Fig. 3, and the good agreement forms thecentral result of our work: the evolution of the electronicconductivity of C6Yb is approximately exponential withapplied ﬁeld, providing the ﬁrst veriﬁcation of s-wavesuperconductivity in intercalated graphite using a bulkthermodynamic technique.At still higher ﬁelds, the conductivity rolls oﬀ andeventually saturates as the sample enters the normalstate. From the data in Fig. 3 we estimate this to occurat H ≃ 0.12 T, in good agreement with estimates of Hc2= 0.11 T from magnetization measurements [5]. WithHc2( c) = 0.12 T we estimate ξab ≃ 525 ˚ A∼ ℓe, whichplaces C6Yb in the dirty limit. This observation is con-sistent with the fact that the rise in conductivity withﬁeld is not as dramatic as in clean Nb [23], but closelyresembles that observed in dirty limit Nb [24] and metalalloy superconductors [25, 26].In addition to conﬁrming s-wave superconductivity,the activated behaviour observed in Fig. 3 rules out thescenario of multi-gap superconductivity originally pro-posed for the GIC’s [11]. In Fig. 4 we compare C6Ybto other type-II superconductors by plotting the normal-ized value of κ0/T versus applied ﬁeld. The similaritybetween C6Yb and the alloyed superconductor InBi withTc = 4.0 K and Hc2 = 0.07 T [25] is striking. Bothcurves are exponential with ﬁeld at low H, crossing overto a roughly linear behaviour closer to Hc2 as expectedfor s-wave superconductors in the dirty limit [28]. Theclean limit case observed in pure Nb [23] is shown forcontrast. These three curves are very diﬀerent from thebehaviour of the archetypal multi-band superconductorMgB2 [27], or the archetypal d-wave superconductor Tl-2201 [18].In the multiband scenario, gaps of diﬀerent magnitudesare associated with the π and σ bands. Such a situation40.0 0.5 1.00.00.51.0NbInBiC6YbMgB2Tl−2201H / Hc2[ k0/T ] / [ kN/T ]FIG. 4: The normalized residual linear term κ0/T of sampleB plotted as a function of H/Hc2, with the small contribu-tion from graphite impurities subtracted oﬀ. For comparison,similar low-temperature data for the clean s-wave supercon-ductor Nb [23], the dirty s-wave superconducting alloy InBi[25], the multi-gap superconductor MgB2 [27] and an over-doped sample of the d-wave superconductor Tl-2201 [18] areplotted alongside.is expected for instance when one band has strong pair-ing and induces superconductivity in the other by Cooperpair tunnelling [29], or when electron-phonon coupling issigniﬁcantly diﬀerent for diﬀerent bands. Applying a ﬁeldrapidly delocalizes quasiparticles states conﬁned withinthe vortices associated with the smaller gap band, whilethose states associated with the larger gap band delocal-ize more slowly. This gives rise to the rapid increase inconductivity at low ﬁelds and relatively ﬂat dependenceat higher ﬁelds [30] evident for in the MgB2 [27] datashown in Fig. 4. Our own data thus allows us to ruleout any sizable diﬀerence between the size of the gaps as-sociated with each band in C6Yb, and suggests a singlegap energy scale for the electrons as in the conventionals-wave scenario.In summary we have used bulk measurements of thethermal conductivity κ to deﬁnitively establish s-wavesuperconductivity in C6Yb, and rule out an order pa-rameter with nodes. The activated behaviour of κ0/Talso strongly suggests that the pairing state is isotropic,very similar to elementary type-II superconductors in thedirty limit. It will interesting to conﬁrm these results onother members of the intercalate family with complemen-tary techniques, although it seems likely that other GICswill share similar superconducting properties.We would like to thank Robert Smith, Sibel ¨ Ozcan andGil Lonzarich for useful discussions. M.S. and N.D-L ac-knowledge support from an NSERC of Canada postdoc-toral fellowship, and L.T. acknowledges support from aCanada Research Chair. This research was funded byEPSRC, NSERC and the CIAR.[1] N. Hannay et al., Phys. Rev. Lett. 14, 225 (1965).[2] Y. Koike et al., Physica B+C 99, 503 (1980).[3] A. Hebard et al., Nature 350, 600 (1991).[4] E. Ekimov et al., Nature 428, 542 (2004).[5] T. E. Weller et al., Nature Physics 1, 39 (2005).[6] N. Emery et al., Phys. Rev. Lett. 95, 087003 (2005).[7] G. Csanyi et al., Nature Physics 1, 42 (2005).[8] I.I. Mazin, Phys. Rev. Lett. 95, 227001 (2005).[9] I.I. Mazin and S. Molodtsov, Phys. Rev. B 72, 172504(2005).[10] M. Calandra and F. Mauri, Phys. Rev. Lett. 95, 237002(2005).[11] R. Al-Jishi, Phys. Rev. B 28, 112 (1983).[12] R. A. Jishi and M. S. Dresselhaus, Phys. Rev. B 45,12465 (1992).[13] P. Konsin and B. Sorkin, Supercond. Sci. Technol. 17,1472 (2004).[14] G. Lamura et al., Phys. Rev. Lett. 96, 107008 (2006).[15] It is probable that all or most of this magnetoresistancearises from small inclusions of pure graphite, estimate tomake up a few percent of our sample in the followingparagraphs.[16] X. Du, S. Tsai, D. Maslov, and A. Hebard, Phys. Rev.Lett. 94, 166601 (2005).[17] A.C. Durst and P.A. Lee, Phys. Rev. B 62, 1270 (2000).[18] C. Proust, E. Boaknin, R.W. Hill, L. Taillefer, and A.P.Mackenzie, Phys. Rev. Lett. 89, 147003 (2002).[19] D. Hawthorn et al., cond-mat/0502273 (2005).[20] D. Morelli and C. Uher, Phys. Rev. B. 30, R1080 (1984).[21] W. Vinen et al., Physica (Amsterdam) 55A, 94 (1971).[22] C. Caroli, P. DeGennes, and J. Matricon, Phys. Lett. 9,307 (1964).[23] J. Lowell and J. Sousa, J. Low. Temp. Phys. 3, 65 (1970).[24] S. Wasim and N. Zebouni, Phys. Rev. 187, 539 (1969).[25] J. Willis and D. Ginsberg, Phys. Rev. B 14, 1916 (1976).[26] R. Parks, F. Zumsteg, and J. Mochel, Phys. Rev. Lett.18, 47 (1967).[27] A. V. Sologubenko, J. Jun, S. Kazakov, J. Karpinski, andH. Ott, Phys. Rev. B 66, 014504 (2002).[28] C. Caroli and M. Cyrot, Phys. Kondens. Mater. 4, 285(1965).[29] N. Nakai, M. Ichioka, and K. Machida, J. Phys. Soc. Jpn.71, 23 (2002).[30] H. Kusononse, T.M. Rice, and M. Sigrist, Phys. Rev. B66, 214503 (2002).

Bulk evidence for single-gap s-wave superconductivity in the intercalated graphite superconductor C_{6}Yb

Bulk evidence for single-gap s-wave superconductivity in the
  intercalated graphite superconductor C$_6$Yb

Bulk evidence for single-gap s-wave superconductivity in the intercalated graphite superconductor C $_6$ Yb

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UCL Discovery

Bulk evidence for single-gap s-wave superconductivity in the intercalated graphite superconductor C6_66​Yb

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UCL Discovery

Bulk evidence for single-gap s-wave superconductivity in the intercalated graphite superconductor C $_6$ Yb