Unresolved problems in superconductivity of CaC6
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trons play little role in the electronic structure at the Fermilevel. Upon the discovery of superconductivity in CaC6Csanyi et al. , comparing band structures of supercon-
discovered a correlation between the occupancy of the so-
more general reasons to believe that neither of the twoproposed electronic mechanisms, Ginzburgs sandwichsuperconductivity and 2D acoustic plasmon mechanism,can be operative in these compounds. We will discuss thisbriey later in the paper and refer the reader for detailsto Ref. .
* Corresponding author. Tel.: +1 202 767 6990; fax: +1 202 404 7546.E-mail address: email@example.com (I.I. Mazin).
Physica C 460462 (200The fact that graphite, a zero-gap semiconductor, be-comes superconducting upon intercalation, has been knownfor some decades already . However, till recently  thecritical temperature was not higher than a few Kelvins.Therefore, the discovery of superconductivity in YbC6 at6.5 K, and, a year later, in CaC6  at 11.5 K, came as a sub-stantial surprise.
Initially, the fact that Yb often exhibits intermediatevalent states inspired speculations of a superconductivitymediated by valence uctuations. It was found, however, by both experimental and theoretical investigationsthat Yb remains divalent in this compound and its f-elec-
called interlayer band and the appearance (and, to someextent, critical temperature) of superconductivity.
This undisputable correlation led Csanyi et al. to a log-ical conjecture that this band (its character and formationwill be discussed below) must play some important rolein superconductivity. However, their next conjecture, thatthis band does not couple to phonons, was not really basedon any computational or experimental fact and turned outto be incorrect . This fact to a large extent rendersmoot the discussion initiated by the exciting suggestion that the superconductivity in AC6 (A = Yb, Ca) is ofelectronic origin, but it is worth noting that there are evenWe discuss the current status of the theory of the high-temperature superconductivity in intercalated graphites YbC6 and CaC6. Weemphasize that while the general picture of conventional, phonon-driven superconductivity has already emerged and is generallyaccepted, there are still interesting problems with this picture, such as weak coupling regime inferred from specic heat suggesting cou-pling exclusively with high-energy carbon phonons coming in direct contradiction with the isotope eect measurements in turn suggestingcoupling exclusively with the low-energy intercalant modes. At the same time, the rst principle calculations, while explaining Tc, con-tradict both the experiments above by predicting equal coupling with both groups of phonons. 2007 Elsevier B.V. All rights reserved.
Keywords: Superconductivity; Intercalated graphite
1. Historical introduction ducting and non-superconducting intercalated graphites,Unresolved problems in s
I.I. Mazin a,*, L. Boeri b, O.V. DolgoM. Giantomassi e
a Center for Computational Materials Science, Navb Max Planck Institute Fur Festk
c Twente University, Faculty of Sciencd INFM and Dipartimento di
e Unite PCPM, Universite Catholique
Available onlin0921-4534/$ - see front matter 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physc.2007.03.276perconductivity of CaC6b, A.A. Golubov c, G.B. Bachelet d,.K. Andersen b
esearch Laboratory, Washington, DC 20375, USA
erforschung, Stuttgart, Germany
d Technology, Enschede, Netherlands
ca, La Sapienza, Roma, Italy
Louvain, Louvain-la-Neuve, Belgium
0 March 2007
the NFEB electrons are localized in the intercalant plane(even without an intercalant!), and therefore the NFEBshould rather be sensible to any charge redistribution inthis plane (think intercalant phonons), which it indeed is,as was rst pointed out in Ref.  and conrmed in Refs.[6,7,15].
Actual linear response calculations [6,7] have producedan interesting and consistent picture: there are three dis-tinctive groups of modes, one at x 10 meV, another atx 60 meV, and the third at x 170 meV, contribut-ing, respectively, 0.4, 0.35, and 0.10 to the total
Fig. 1. Left panel: Band structure of CaC6, with the NFEB emphasized inblue, and carbon p bands in red. Right panel: the same for a non-intercalated graphite. (For interpretation of the references in colour in thisgure legend, the reader is referred to the web version of the article.)
Fig. 2. Isocontours of the amplitude of the Wannier function (red:positive, blue: negative) corresponding to the NFEB. Left panel: CaC6,Right panel: C6. (For interpretation of the references in colour in thisgure legend, the reader is referred to the web version of the article.)
CThe next step in unraveling the mechanism of supercon-ductivity in AC6 was taken by one of us in Ref. , where itwas pointed out that the square root of the mass ratio ofthe intercalants, Yb and Ca, is just 15% larger than theratio of the critical temperatures. At that moment it hadbeen reported that the crystal structure of the two com-pounds was identical. This, together with the establishedvalency 2 of Yb in YbC6, led to the conclusion  thatthe electronic structures of the two compounds are extre-mely similar, and therefore the dierence in their criticaltemperatures must be mostly due to the dierence in thephonon frequency of the intercalant atoms (which, inciden-tally, was in line with the trend discovered by Csanyi et al.that lling of the interlayer band correlates with supercon-ductivity). Thus, a prediction was made that the isotopeeect on Ca should not be much reduced from its idealvalue of 0.5 (15% reduction being a natural rule-of-thumbguess).
However, Emery et al. turned the next page in the storyof CaC6  by showing that its crystal structure is actuallydierent from that of YbC6, not much but enough to ren-der Mazins estimate  inaccurate. Indeed, the rst princi-ple calculations of Mauri and Calandra  and of Boeriet al.  showed that the intercalant modes do not dominatethe electronphonon coupling, but contribute to it about asmuch as the graphite modes.
These calculations predicted a moderately strong cou-pling (k 0.85). Solving the Eliashberg equations withthe calculated a2F(x), with a reasonable l = 0.12, givesthe experimental Tc. While the conclusion that supercon-ductivity in AC6 is conventional, s-wave, phonon-medi-ated, has found full support in numerous experiments, asdiscussed below, there are still alarming quantitative dis-crepancies between the theory and the experiment, as wellas among dierent experiments.
2. Electrons and phonons in CaC6
As discussed above, the main feature that distinguishesthe superconducting intercalated graphites is the appear-ance at the Fermi level of a three-dimensional nearly-free-electron band (NFEB) that some authors (e.g., Ref.) call a free electron band, and others (e.g., Ref. ) anintercalant band, yet others an interlayer band (e.g., and references therein). The former refer to the fact thatone can remove Ca from the system and still identify theNFEB, but much higher in energy than in CaC6. The latterargue that one can remove carbon instead, leaving a some-what expanded along 111 metallic Ca and still see the sameband! Of course, the root of the controversy is in the factthat pure Ca is a nearly-free-electron metal and its sp bandsare rather similar to the free electron bands (Figs. 1 and 2).
From Fig. 1 one can see that the NFEB is present inboth cases. From Fig. 2 it is obvious that there are never-theless important dierences: in CaC6, calcium d(z
I.I. Mazin et al. / Physicaorbitals participate in the formation of this band, as wellas carbon p(z). Most importantly, Fig. 2 clearly shows that460462 (2007) 116120 117coupling constant k (Fig. 3). These three groups aremainly composed of the out-of-plane Ca and in-plane C
- Fig. 3. (a) Calculated phonon dispersion and (b) density of states of CaC6for P = 0 and P = 100 kbar, (c) Eliashberg function and frequency-dependent electronphonon coupling constant kx 2 Rx
After this paper had been submitted, density functionalcalculations suggested a moderately anisotropic orderparameter have been published  that may be capableof resolving the specic heat problem. The large Ca isotopeeect still remains a mystery, though.
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Cof the order of several eV, i.e., 2D acoustic phonons are notpresent in the system.
2.3. Multiband superconductivity
Although currently the interest to multigap supercon-ductivity is high, thanks to MgB2, the idea of a two-gapsuperconductivity in intercalated graphites is very old. The problem here is that, as Fig. 4 shows, in this caseone can speak, at best, about anisotropic superconductivitywith an order parameter radically changing over one andthe same Fermi surface sheet. While not impossible, itmakes a two-gap superconductivity less likely. Indeed,experimental evidence is mounting in favor of one isotropicorder parameter .
3. Unsolved problems
At rst glance, it may seem that the superconductivity inAC6 is fully explained in terms of the conventional, stan-dard, isotropic BCS theory. Unfortunately, this is not thecase. Below we list the questions that still await answers.
3.1. Critical eld
As pointed out in Refs. [15,18], Hc2(T) remains lineardown to 1 K, which contradicts the standard BCS model.In principle, a very strong coupling can render a linearHc2(T) ; however, we have calculated Hc2(T) using theLDA a2F(x) and found that sublinear behavior startsalready at T < 4 K. A more likely explanation has to dowith the Fermi velocity anisotropy, which can lead to a lin-ear Hc2 even in a cubic superconductor like K3C60 .A good rule-of-thumb is that if half of the Fermi surfacehas a velocity twice larger than the other half, Hc2(T)is linear [e.g., according to the cubic formula H c20=HBCSc2 0 hv2Fi=hexpln v2Fi]. However, this explanationneeds a quantitative verication.
3.2. Specic heat
An analysis of C(T) below Tc in the framework of thea-model  yields a = 1.776, within 1% of the BCS value.C/T renormalization is 1.6, to be compared to the calcu-lated [6,7] k = 0.85. This seems like a good agreement,but, as Fig. 5 shows, the calculated a2F(x) produces strongcoupling eects in C(T) well beyond the experimental error.We have also checked that the disagreement is stableagainst the shape of a2F(x), by repeating the calculationsusing the same k = 0.85, but condensing the a2F(x) toone Einstein mode. The shape of DC(T)/T did change,but not enough to explain the experiment.
3.3. Isotope eect
I.I. Mazin et al. / PhysicaHinks et al.  have measured the isotope eect on Caand found it to be 0.5, as opposed to 0.25 predicted by thecalculations . Since all Ca modes are rather soft, cou-pling exclusively with Ca phonons must necessarily be verystrong, in disagreement with the specic heat data suggest-ing a typical weak coupling regime, and with other indica-tions of weak coupling.
Thus, the main challenge regarding the superconductiv-ity in AC6 is to reconcile the evidence for weak coupling(i.e., dominance of high-energy C modes)  with evidencefor the dominance of the intercalant modes (i.e., strongcoupling)  and with the computations [6,7] and experi-ment  (this experiment showed that both types of modesare needed to quantitatively explain the temperaturedependence of resistivity, and that with pressure the cou-pling with all modes remains constant, while the Ca modessoften until becoming unstable) predicting intermediatecoupling with both C and Ca equally involved.
0 4 8 10 12
Calculated using a 2F(w) of Boeri et al
Experiment (Kim et al)
T (K)2 6
Fig. 5. Deviation of the calculated (strong coupling) specic heat from theexperiment .
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Unresolved problems in superconductivity of CaC6Historical introductionElectrons and phonons in CaC6Ginzburg ' s sandwichAcoustic plasmonsMultiband superconductivity
Unsolved problemsCritical fieldSpecific heatIsotope effect