Theory of Superconductivity in Graphite Intercalation Compounds
Yasutami Takada1, ∗1

arXiv:1601.02753v1 [cond-mat.supr-con] 12 Jan 2016

Institute for Solid State Physics, University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, JapanOn the basis of the model that was successfully applied to KC8 , RbC8 , and CsC8 in 1982, wehave calculated the superconducting transition temperature Tc for CaC6 and YbC6 to find that thesame model reproduces the observed Tc in those compounds as well, indicating that it is a standard
model for superconductivity in the graphite intercalation compounds with Tc ranging over threeorders of magnitude. Further enhancement of Tc well beyond 10 K is also predicted. The present
method for calculating Tc from first principles is compared with that in the density functional theory
for superconductors, with paying attention to the feature of determining Tc without resort to theconcept of the Coulomb pseudopotential µ∗ .
PACS numbers: 74.70.Wz,74.20.-z,74.20.Pq

I.

INTRODUCTIONA.

Crystal Structure

For many decades, graphite intercalation compounds(GICs) have been investigated from the viewpoints ofphysics, chemistry, materials science, and engineering (ortechnological) applications1–4 . Among various kinds of
GICs, special attention has been paid to the first-stage
metal compounds, partly because superconductivity isobserved mostly in this class of GICs, the chemical formula of which is written as M Cx , where M represents
either an alkali atom (such as Li, K, Rb, and Cs) or an
alkaline-earth atom (such as Ca, Sr, Ba, and Yb) andx is either 2, 6, or 8. The crystal structure of M Cx isshown in Fig. 1(a), in which the metal atom M occupies
the same spot in the framework of a honeycomb lattice
at every (x/2) layers of carbon atoms.

FIG. 1: (a) Crystal structure of M Cx ix = 2, 6, 8). The case ofx = 6 is illustrated here, in which the metal atoms, M s, are arranged in a rhombohedral structure with the αβγ stacking sequence, implying that M occupies the same spot in the framework of a graphene lattice at every three layers (or at the distance of 3d with d the distance between the adjacent graphite
layers). (b) Superconducting transition temperature Tc observed in the first-stage alkali- and alkaline-earth-intercalated
graphites plotted as a function of d.

B.

Superconductivity

The first discovery of superconductivity in GICs wasmade in KC8 with the superconducting transition temperature Tc of 0.15K in 19655 . In pursuit of higher Tc ,various GICs were synthesized, mostly working with thealkali metals and alkali-metal amalgams as intercalants,from the late 1970s to the early 1990s6–13 , but only alimited success was achieved at that time; the highest
attained Tc was around 2-5K in the last century. For
example, it is 1.9K in LiC2 14 .
A breakthough occurred in 2005 when Tc went up to
11.5K in CaC6 15,16 (and even to 15.4K under pressures
up to 7.5GPa17 ). In other alkaline-earth GICs, the valuesof Tc are 6.5K, 1.65K, and 0.065K for YbC6 15 , SrC6 18 ,and BaC6 19 , respectively, as indicated in Fig. 1(b). Since
then, very intensive experimental studies have been madein those and related compounds16,18,20,21 . Theoretical studies have also been performed mainly by making state-of-the-art first-principles calculations of theelectron-phonon coupling constant λ to account for theobserved value of Tc for each individual superconductor22–26 . Those experimental/theoretical works haveelucidated that, although there are some anisotropic
features in the superconducting gap, the conventional
phonon-driven mechanism to bring about s-wave superconductivity applies to those compounds. This pictureof superconductivity is confirmed by, for example, theobservation of the Ca isotope effect with its exponent
α = 0.50, the typical Bardeen-Cooper-Schrieffer (BCS)
value27 .
C.

Central Issues

In spite of all those efforts and the existence of sucha generally accepted picture, there remain several very
important and fundamental questions:
1. Can we understand the mechanism of superconductivity in both alkali GICs with Tc in the range
0.01 − 1.0K and alkaline-earth GICs with Tc typi-

2cally in the range 1 − 10K from a unified point ofview? In other words, is there any standard model
for superconductivity in GICs with Tc ranging overthree orders of magnitude?
2. What is the actual reason why Tc is enhanced so
abruptly (or by about a hundred times) by justsubstituting K by Ca the atomic mass of which isalmost the same as that of K? In terms of the standard model, what are the key controlling physicalparameters to bring about this huge enhancementof Tc ? This change of Tc from KC8 to CaC6 is
probably the most important issue in exploring superconductivity across the entire family of GICs.
3. Is there any possibility to make a further enhancement of Tc in GICs? If possible, what is the optimum value of Tc expected in the standard modeland what kind of atoms should be intercalated torealize the optimum Tc in actual GICs?
4. What is the physical reason why BaC6 provides so
low Tc (=0.065K) experimentally, compared withother alkaline-earth GICs and also with Tc = 0.23K
in the conventional Eliashberg theory25 ? This issue
may not be important in obtaining high-Tc superconductors, but physically it is important enoughin comprehensively understanding the mechanismof superconductivity in GICs.
In order to provide reliable answers to the above questions, it is indispensable to make a first-principles calculation of Tc with sufficient accuracy and predictive
power. Several years ago, such a calculation was completed by the present author and on the basis of thecalculation, some interesting predictions has been proposed28,29 . The present chapter not only reports somedetails of this work on the superconducting mechanism
in GICs but also makes a brief summary of the current
status of the theories for first-principles calculations of
Tc .

D.

Organization of This Chapter

This chapter is organized as follows: In Secs. 2-4, a critical review of the theories for quantitative calculations of
Tc is given. More specifically, we make comments on thetheories based on the McMillan’s or the Allen-Dynes’ formula employing the concept of the Coulomb pseudopotential µ∗ in Sec. 2. In Sec. 3 we explain the theory on thelevel of the so-called G0 W0 approximation, in which it isa very important merit that Tc can be obtained without
using µ∗ . The same merit can be enjoyed in the density functional theory for superconductors, which will be
addressed in Sec. 4. In Secs. 5-8, a review on superconductivity in GICs is given; starting with a summary of
the experimental works in Sec. 5, a standard model for
considering the mechanism of superconductivity in GICs

is introduced in Sec. 6. In Sec. 7, the calculated resultsof Tc are given for the alkaline-earth GICs and they arecompared with the experimental results. The predictionof the optimum Tc is given in Sec. 8. Finally in Sec. 9,
the conclusion of this chapter is given, together with someperspectives on the researches in this and related fields
in the future.

II.

FIRST-PRINCIPLES CALCULATION OF Tc
A.

Goal of the Problem

It would be one of the ultimate goals in the enterpriseof condensed matter theory to make a reliable predictionof Tc only through the information on constituent elements of a superconductor in consideration. A less ambitious yet very important goal is to make an accurate evaluation of Tc directly from a microscopic (model) Hamiltonian pertinent to the superconductor. If we could findthe dependence of Tc on the parameters specifying themodel Hamiltonian, we could obtain a deep insight into
the mechanism of superconductivity or the competition
between the attractive and the repulsive interactions between electrons. Accumulation of such information might
pave the way to the synthesis of a room-temperature superconductor, a big dream in materials science. From
this perspective, a continuous effort has been made fora long time to develop a good theory for first-principles
calculations of Tc , starting with a microscopic Hamiltonian.

B.

McMillan’s and Allen-Dynes’ Formulas for Tc

In the phonon mechanism, for example, there has beena rather successful framework for this purpose, known
as the McMillan’s formula30 or its revised version (theAllen-Dynes’ formula)31–33 , both of which are derived
from the Eliashberg theory of superconductivity34 . In
this framework, the task of a microscopic calculation of
Tc is reduced to the evaluation of the so-called Eliashberg function α2 F (ω) from first principles; the function
α2 F (ω) enables us to obtain both the electron-phonon
coupling constant λ and the average phonon energy ω0 ,through which we can make a first-principles predictionof Tc with an additional introduction of a phenomenological parameter µ∗ (the Coulomb pseudopotential35 )in order to roughly estimate the effect of the Coulomb
repulsion between electrons on Tc .
At present this framework is usually regarded as the
canonical one for making a first-principles prediction of
Tc . In fact, the superconducting mechanism of many(so-called weakly-correlated) superconductors is believed
to be clarified by using this framework, whereby the key
phonon modes to bring about superconductivity are identified. We can point out that superconductivity in MgB2

3
with Tc = 39K is a good recent example36–39 to illustrate the power of this framework. The case of CaC6
has also been investigated along this line of theoreticalstudies22,24 .

C.

Coulomb Pseudopotential µ∗

Nevertheless, this framework is not considered to bevery satisfactory, primarily because a phenomenologicalparameter µ∗ is included in the theory. Actually, it cannot be regarded as the method of predicting Tc in thetrue sense of the word, if the parameter µ∗ is determinedso as to reproduce the observed Tc . Besides, as long as we
employ µ∗ to avoid a serious investigation of the effectsof the Coulomb repulsion on superconductivity, we cannot apply this framework to strongly-correlated superconductors. Even in weakly- or moderately-correlated
superconductors, this framework cannot tell anything
about superconductivity originating from the Coulomb
repulsion via charge and/or spin fluctuations (namely,
the electronic mechanism including the plasmon mechanism40,41 ). Furthermore, in this framework, we cannot
investigate the competition or the coexistence (or eventhe mutual enhancement due to the quantum-mechanicalinterference effect) between the phonon and the electronic mechanisms.
The validity of the concept of µ∗ is closely connectedwith that of the Eliashberg theory itself; the theory isvalid only when the Fermi energy of the superconducting
electronic system, EF , is very much larger than ω0 . Under the condition of EF ≫ ω0 , the dynamic response time
for the phonon-mediated attraction ω0−1 is much slower
than that for the Coulomb repulsion EF−1 , precluding any
possible interference effects between two interactions, sothat physically it is very reasonable to separate them.
After this separation, the Coulomb part (which was notconsidered to play a positive role in the Cooper-pair formation) has been simply treated in terms of µ∗ . Thus,
for the purpose of searching for some positive role of the
Coulomb repulsion, the concept of µ∗ is irrelevant fromthe outset.

D.

Vertex Corrections and Dynamic Screening

Incidentally, in some recently discovered superconductors in the phonon mechanism such as the alkali-doped
fullerenes with Tc = 18 − 38K42–45 , the condition ofEF ≫ ω0 is violated, necessitating to include the vertex corrections in calculating the phonon-mediated attractive interaction46 . Then, it is by no means clear totreat the overall effect of various phonons in terms of
the sum of the contribution from each phonon, implyingthat the Eliashberg function α2 F (ω) is not enough toproperly describe the attraction because of possible interference effects among virtually-excited phonons. As aconsequence, λ will not be simply the sum of λi the con-tribution from the ith phonon, unless λi is small enoughto validate the whole calculation in lowest-order perturbation.
In the case of EF ≈ ω0 , another complication occurs intreating the screening effect of the conduction electrons.
In the usual first-principles calculation scheme, the static
screening is assumed in calculating α2 F (ω), but it does
not reflect the actual screening process working during
the formation of Cooper pairs.

E.

Ideal Calculation Scheme

In order to unambiguously solve this problem of screening, we may imagine a following ideal calculation scheme
for Tc : In the first step, we calculate the microscopic dynamical electron-electron effective interaction V in thewhole momentum and energy space. This V is assumed
to contain both the Coulomb repulsion and the phononmediated attraction on the same footing. Then in thesecond step, we obtain Tc directly from this V with simultaneously determining the gap function in the whole
momentum and energy space, reflecting the behavior of
V . If this scheme were developed, we could not only calculate Tc from first principles without resort to µ∗ but
also correctly discuss the competition, coexistence, andmutual enhancement between the phonon and the electronic mechanisms.

III.

CALCULATION OF Tc IN THE G0 W0
APPROXIMATIONA.

Formulation

Although it is along the royal road in the project of obtaining a reliable method for predicting Tc from first principles, this ideal calculation scheme is extremely difficult
to achieve in actual situations, because all the difficulties
in the quantum-mechanical many-body problem are associated with it. About three decades ago, the present
author, who was a graduate student at that time, wasstruggling with developing such a scheme without perceiving much of the difficulties intrinsic to the many-bodyproblem. After a year-long struggle, he managed to propose a rather general scheme to evaluate Tc directly from
V without introducing the concept of µ∗40 , though it was
still at the stage far from the ideal scheme.
In a broad sense, this scheme may be called an approach from the weak-coupling limit, corresponding to
the G0 W0 approximation or the one-shot GW approximation in the terminology of the present-day firstprinciples calculation community. In the same terminology, by the way, the Eliashberg theory corresponds to the
GW approximation with respect to the phonon-mediatedattractive interaction between electrons.
Let us explain this G0 W0 scheme here47 . For simplicity, imagine the three-dimensional (3D) electron gas

4
in which an electron is specified by momentum p andspin σ. If we write the electron annihilation operator by
cpσ , we can define the abnormal thermal Green’s function
F (p, iωp ) at temperature T by
F (p, iωp ) = −

Z 1/T
0

dτ eiωp τ hTτ cp↑ (τ )c−p↓ i,(1)with ωp the fermion Matsubara frequency. At T = Tc
where the second-order phase transition occurs, this function satisfies the following exact gap equation:
F (p, iωp ) = −G(p, iωp )G(−p, −iωp )XX
˜ p′ ; iωp , iωp′ )F (p′ , iωp′ ),
×Tc
I(p,

(2)

ω p ′ p′where G(p, iωp ) is the normal Green’s function and˜ p′ ; iωp , iωp′ ) is the irreducible electron-electron efI(p,fective interaction.
Now, in the spirit of the G0 W0 approximation, we replace G(p, iωp ) by the bare one G0 (p, iωp ) ≡ (iωp −εp )−1
in Eq. (2), where εp (= p2 /2m∗ − µ) is the bare oneelectron dispersion relation with m∗ the band mass andµ the chemical potential. We will also consider the case˜ p′ ; iωp , iωp′ ) is well approximated as a funcin which I(p,tion of only the variables (p−p′ , iωp −iωp′ ) to write
˜ p′ ; iωp , iωp′ ) = V (p−p′ , iωp −iωp′ ),
I(p,

(3)

just like the effective interaction in the random-phaseapproximation (RPA), though we do not intend to confine ourselves to the RPA at this point. By substituting
Eq. (3) into Eq. (2) and making an analytic continuationon the ω plane to transform F (p, iωp ) to the retarded
function F R (p, ω) on the real-ω axis, we get a gap equation for F R (p, ω). Then, by taking the imaginary partsin both sides of the gap equation and integrating over theω variable, we finally obtain an equation depending only
on the momentum variable p. Concretely, the equation
can be cast into the following BCS-type gap equation:∆p = −

X ∆p′
p′

2εp′

tanh

ε p′
Kp,p′ ,
2Tc

(4)

where the gap function ∆p and the pairing interaction
Kp,p′ are, respectively, defined by∆p ≡ 2|εp |

Z ∞
0

dω
Im F R (p, ω),
π

(5)

and
Kp,p′ =Z ∞0

|εp | + |εp′ |2
V (p−p′ , iΩ) . (6)dΩ 2πΩ + (|εp | + |εp′ |)2

With use of Kp,p′ thus calculated, we can determine Tc
as an eigenvalue of Eq. (4).

B.

Comments on the Formulation

Five comments are in order on this framework:i) Based on Eqs. (4) and (6), we can obtain Tc directlyfrom the microscopic one-electron dispersion relation εp and the effective electron-electron interaction V (q, iΩ) with no need to separate the phononmediated attraction from the Coulomb repulsion.
ii) In spite of the similarity of Eq. (4) to the BCS gap
equation, artificial cutoffs involved in constructing
the BCS model are avoided in the present scheme;
natural cutoffs are automatically introduced by thecalculation of Kp,p′ defined in Eq. (6).
iii) Except for the spin-singlet pairing, no assumption
is made on the dependence of the gap function on
angular valuables in deriving Eq. (4), so that thisgap equation can treat any kind of anisotropy in
the gap function, indicating that it can be applied
to s-wave, d-wave, · · · , and even their mixture like(s+d)-wave superconductors.
iv) As can be seen by its definition, the gap function∆p in Eq. (5) does not correspond to the physical energy gap except in the weak-coupling region.
Similary, Kp,p′ is not a physical entity. Both quantities are introduced for the mathematical convenience so as to make Tc invariant in transforming
Eq. (2) into Eq. (4). The key point here is that weneed not solve the full gap equation (2) but much
simpler one (4) in order to obtain Tc in Eq. (2). Of
course, if we want to know the physical gap function rather than ∆p to compare with experiment,
we need to solve the full gap equation, Eq. (2), with
Tc determined by Eq. (4).
v) Historically, Cohen was the first to evaluate Tc
in degenerate semiconductors on the level of theG0 W0 approximation48,49 . Unfortunately the pairing interaction is not correctly derived in his theory,
as explicitly pointed out by the present author50
who, instead, has succeeded in obtaining the correct pairing interaction40 by consulting the pertinent work of Kirzhnits et al.51 .
C.

Assessment: Application to SrTiO3

In order to assess the quality of this basic frameworkof calculating Tc from first principles, we have applied it
to SrTiO3 and compared the results with experiments50 .
This material is an insulator and exhibits ferroelectricity
under a uniaxial stress of about 1.6kbar along the [100]direction, but it turns into an n-type semiconductor byeither Nb doping or oxygen deficiency, whereby the conduction electrons are introduced in the 3d band of Ti
around the Γ point with the band mass of m∗ ≈ 1.8me(me : the mass of a free electron). At low temperatures,

5superconductivity appears and the observed Tc shows interesting features; Tc depends strongly on the electron
concentration n and it is optimized with Tc ≈ 0.3K atn ≈ 1020 cm−3 . Its dependence on the pressure is unsual;Tc decreases rather rapidly with hydrostatic pressures,but it increases with the [100] uniaxial stress.

FIG. 2: (a) Electron density dependence of Tc in SrTiO3 and(b) pressure dependence of Tc 50 . Both uniaxial stress along
either [100] or [110] direction and hydrostatic pressure (H.P.)
are considered. The deviation of Tc , ∆Tc , is plotted as a function of stress in units of kbar. For comparison, corresponding
experimental results are also shown, together with the resultsof the transverse polar phonon energy ωt in units of cm−1 .

Taking those situations into account, we have assumed
that the superconductivity is brought about by the polarcoupling phonons associated with the stress-induced ferroelectric phase transition. Then we have calculated
the effective electron-electron interaction V (q, iΩ) in the
RPA, in which the long-range attraction induced by thevirtual exchange of polar-coupled phonons is included
with the long-range Coulomb repulsion on the same footing. By substituting this V (q, iΩ) into Eq. (6), we haveobtained Tc directly from a microscopic model and theresults of Tc are in surprizingly good quantitative agreement with experiment, as shown in Fig. 2. Recently, afurther experimental study on superconductivity in thiselectron-doped SrTiO3 was made to confirm the above
results, together with the value for the effective mass ofm∗ = (1.82 ± 0.05)me 52,53 . This success indicates thatthe present basic framework including the adoption of the
RPA is very useful at least in the polar-coupled phononmechanism.
IV.

DENSITY FUNCTIONAL THEORY FOR
SUPERCONDUCTORSA.

Hohenberg-Kohn-Sham Theorem

Recently much attention has been paid to an extensionof the density functional theory (DFT) to treat superconductivity, mainly because it provides another scheme forfirst-principles calculations of Tc without resort to µ∗ .
We shall make a very brief review of it in this section.

It is stated in the basic theorem in DFT that all thephysics of an interacting electron sysytem is uniquely determined, once its electronic density in the ground staten(r) is specified. This Hohenberg-Kohn theorem54 implies that every physical quantity including the exchangecorrelation energy Fxc may be considered as a uniquefunctional of n(r). The density n(r) itself can be determined by solving the ground-state electronic density of
the corresponding noninteracting reference system thatis stipulated in terms of the Kohn-Sham (KS) equation55 . The core quantity in the KS equation is theexchange-correlation potential Vxc (r), which is definedas the functional derivative of Fxc [n(r)] with respect ton(r), namely, Vxc (r) = δFxc [n]/δn(r). It must be notedthat Vxc (r) as well as each one-electronic wavefunction at
ith level with its energy eigenvalue εi in the KS equation
has no physical relevance; they are merely introduced forthe mathematical convenience so as to obtain the exactn(r) in connecting the noninteracting reference systemwith the real many-electron system.
The Hohenberg-Kohn theorem can be applied to theordered ground state as well on the understanding thatthe order parameter itself is regarded as a functional ofn(r). In providing some approximate functional form forFxc [n], however, it would be more convenient to treat
the order parameter as an additional independent variable. For example, in considering the system with somemagnetic order, we usually employ the spin-dependent
scheme in which the fundamental variable is not n(r) but
the spin-decomposed density nσ (r), leading to the spinpolarized exchange-correlation energy functional Fxc [nσ ],based on which the spin-dependent KS equation is formulated.

B.

Gap Equation

Similarly, in treating the superconducting state in theframework of DFT, it is better to construct the energyfunctional with employing both n(r) and the electronpair density χ(r, r ′ )(≡ hΨ↑ (r)Ψ↓ (r ′ )i) as basic variables,
leading to the introduction of the exchange-correlationenergy functional Fxc [n(r), χ(r, r ′ )], where Ψσ (r) is theelectron annihilation operator56,57 . In accordance withthis addition of the order parameter as a fundamental variable to DFT, not only the exchange-correlation
potential Vxc (r) but also the exchange-correlation pairpotential ∆xc (r, r ′ ) = −δFxc [n, χ]/δχ∗ (r, r ′ ) appear in
an extended KS equation, which is found to be written
in the form of the Bogoliubov-de Gennes equation appearing in the usual theory for inhomogeneous superconductors58 . Just as is the case with Vxc (r), ∆xc (r, r ′ ) has
no direct physical meaning, but in principle, if the exact
form of Fxc [n, χ] is known, the solution of the extended
KS equation gives us the exact result for χ(r, r ′ ), containing all the effects of the Coulomb repulsion including
the one usually treated phenomenologically through theconcept of µ∗ . As a result, we can determine the exact

6
Tc by the calculation of the highest temperature below
which a nonzero solution for χ(r, r ′ ) can be found.
In this formulation, we can write the fundamental gap
equation to determine Tc exactly as∆i = −

X ∆j
j

2εj

tanh

εj
Kij ,
2Tc

(7)

where ∆i is the gap function for ith KS level. In just thesame way as its energy eigenvalue εi (which is measured
from the chemical potential), ∆i is not the quantity to
be observed experimentally but just introduced for themathematical convenience so as to obtain the exact Tc
by solving this BCS-type equation, Eq. (7). Similarly,
the pair interaction Kij , defined as the second-functional
derivative of Fxc [n, χ] with respect to χ∗ and χ, has not
any direct physical meaning, either.
We note here the very impressive fact that the final
forms for the two gap equations, Eqs. (4) and (7), are exactly the same, in spite of the fact that they are derived
from quite different foundations and reasoning. We also
note that because of this similarity, we may judge that,as long as Kij is properly chosen, the physics descibedby µ∗ is also included in the framework of DFT for superconductors, at least to the extent that it is includedin the G0 W0 scheme explained in Sec. 3.
C.

Applications

In 2005, this DFT framework was extended to explicitly taking care of the phonon-mediated attractive interaction59 and it has been applied to many superconductors26,60–64 . In order to perform these calculations for
actual superconductors, it is necessary to provide a concrete form for Fxc [n, χ]. In the judgement of the present
author, the presently available form for Fxc [n, χ] contains
the information equivalent to that included in the Eliashberg theory for the part of the phonon-mediated attraction, indicating that no vertex corrections are consideredin this treatment, while for the part of the Coulomb repulsion, it contains only very crude physics; the screening
effect is treated in the Thomas-Fermi static-screening approximation, which is nothing but the result of the RPA
only in the static and the long-wavelength limit, forgetting the detailed dynamical nature of the screening effect.
Mainly for this reason, Tc in the present form of Fxc [n, χ]
is not expected to be very accurate, even though the calculated results for Tc seem to be in good agreement withexpeiment.

D.

Basic Problems

In relation to the above point, it would be appropriate to give a following comment: In the calculations of
the normal-state properties in the local-density approximation (LDA) and generalized gradient approximation(GGA)65 to DFT, we usually anticipate that errors in
the calculated results are of the order of 1eV and 0.3eV
for LDA and GGA, respectively. Those errors are muchlarger than that expected in the calculation of quantum
chemistry (≈ 0.05eV). In DFT for superconductors, calculations of Tc (which is of the order of 0.001eV in general) are done simultaneously with those for the normalstate properties. This implies that the errors anticipatedfor Tc would be very large compared to Tc itself.
We should also point out that the present form forFxc [n, χ] is useless to discuss the electronic mechanisms
like the plasmon and the spin-fluctuation ones, prompting us to improve on the approximate form for Fxc [n, χ].
Very recently, a limited improvement on Fxc [n, χ] wasmade by the inclusion of the contribution from plasmons,leading to better agreement with experiment for Tc 66,67 .
Apart from the functional form, there are also several
problems in the fundamental theory; for example, it is by
no means clear whether the second-functional derivativeof Fxc [n, χ] is a well-defined quantity or not, in just thesame way as we have already experienced in the energygap problem68–70 in semiconductors and insulators.

V.

EXPERIMENT ON SUPERCONDUCTIVITY
IN GICS

From this section, let us get back to the review on superconductivity in GICs. As briefly mentioned in Sec. 1,
the history of the researches on this issue extends morethan four decades. In 1965, the first report of superconductivity was made for KC8 , RbC8 , and CsC8 5 , in which
Tc was not reliably determined; it depended very muchon samples. Subsequent works6–8,10,71–74 confirmed theoccurrence of superconductivity in KC8 with Tc = 0.15K,
but superconductivity did not appear in RbC8 and CsC8
down to 0.09K and 0.06K, respectively. Later works havefound that Tc is actually 26mK for RbC8 3 , but no superconductivity is found in either LiC6 or the secondor higher-stage alkali GICs, though the calculation ofTc based on the McMillan’s formula30 predicted an observable value of Tc even for KC24 75,76 . It seems thatthe usual first-principles calculation of α2 F (ω) tends toprovide an unrealistically large contribution from the intralayer high-energy carbon oscillations to λ. This unfavorable tendency in the calculation of λ seems to prevail
even in CaC6 24 .
The anisotropy of the critical magnetic field Hc2 wasalso a matter of interest, drawing attention of both experimentalists8,10,12,74,77 and theorists78,79 . Note that the
gap function ∆p defined in Eq. (4) has nothing to do withthe anisotropic behavior of Hc2 , though in developing
a phenomenological theory78,79 , some critical comments
were made on the results of ∆p 80 with the assumptionthat the anisotropy in Hc2 should reflect on ∆p .
In search of higher Tc , many attempts have been
made to synthesize new GIC superconductors such as
NaC2 (Tc = 5K)81 , LiC2 (Tc = 1.9K)14 , and alkali-

7metal amalgams like KHgC4 (Tc = 0.73K) and KHgC8(Tc = 1.90K).9,10,72,73,82,83 , but a larger enhancement of
Tc was not achieved until CaC6 was found in 2005 with
Tc = 11.5K15 . Subsequently, many works have been doneon alkaline-earth GIC superconductors16,18,20,21,27,84–87 ,but no one has ever succeeded in synthesizing a new
GIC with Tc larger than 15.4K which was observed in
CaC6 under pressures17 . Thus some new idea seems tobe needed to further enhance Tc . The present author
hopes that the suggestions given in Sec. 8 help experimentalists synthesize a new GIC superconductor with Tc
much higher than 10K.
VI. STANDARD MODEL FOR
SUPERCONDUCTIVITY IN GICS
A.

Characteristic Features of the System

Basically because GICs are not recognized as stronglycorrelated systems, the usual ab initio self-consistentband-structure calculation is very useful in elucidating
the important features of the electronic structures of
GICs in the normal state. According to such calculations,
it is found that there is no essential qualitative difference
between alkali and alkaline-earth GICs (see Fig. 3). The
main common features among these GICs may be summarized in the following way:Z(1−f ) electrons will go to the π and the 3D bands,
respectively.
c) The electrons in the graphite π bands are characterized by the two-dimensional (2D) motion with alinear dispersion relation (known as a Dirac cone in
the case of graphene) on the graphite layer.
d) The dispersion relation of the graphite interlayerband is very similar to that of the 3D free-electrongas, folded into the Brillouin zone of the graphite23 .
Thus its energy level is very high above the Fermi
level in the graphite, because the amplitude of the
wavefunction for this band is small on the carbon
atoms. In M Cx , on the other hand, the cation
M Z+ is located in the interlayer position where
the amplitude of the wavefunctions is large, lowering the energy level of the interlayer band below
the Fermi level. The dispersion of the interlayerband is modified from that of the free-electron gas
because of the hybridization with the orbitals associated with M , but generally it is well approximated by εp = p2 /2m∗ − EF with an appropriate choice of the effective band mass m∗ and the
Fermi energy EF . Here the value of m∗ depends on
M ; in alkali GICs, the hybridization occurs withs-orbitals, allowing us to consider that m∗ = me ,while in alkaline-earth GICs, the hybridization withd-orbitals contributes much, leading to m∗ ≈ 3mein both CaC6 and YbC6 , as revealed by the bandstructure calculation22,24 .
e) The value of f , which determins the branching ratio
Zf : Z(1−f ), can be obtained by the self-consistentband-structure calculation. In KC8 , for example,
it is known that f is around 0.691 . On the other
hand, f is about 0.1624 in CaC6 , making the electron density in the 3D band n increase very much.
This increase in n is easily understood by the factthat the energy level of the interlayer band is much
lower with Ca2+ than with K+ . The concrete numbers for n are 3.5 × 1021 cm−3 and 2.4 × 1022 cm−3
for KC8 and CaC6 , respectively, in which the difference in both d and x is also taking into account.

FIG. 3: (a)Band structure of CaC6 24 . (b)Fermi surface ofKC8 92 . Both materials are characterized by the common feature that the electronic system is composed of the 2D π bandsof graphite and the 3D interlayer band.

a) In M Cx , each intercalant metal atom acts as adonor and changes from a neutral atom M to an
ion M Z+ with valence Z.
b) The valence electrons released from M will transfer either to the graphite π bands or the threedimensional (3D) band composed of the intercalantorbitals and the graphite interlayer states88–90 . We
shall define the factor f as the branching ratio between these two kinds of bands. Namely, Zf andf) As inferred from experiments2,23,80 and also fromthe comparison of Tc calculated for each band80 ,it has been concluded that only the 3D interlayerband is responsible for superconductivity. Notethat LiC6 does not exhibit superconductivity because no carriers are present in the 3D interlayer
band, although the properties of LiC6 are generally very similar to those of other superconducting
GICs in the normal state.
B.

Microscopic Model for Superconductivity

With these common features in mind, we can thinkof a simple model for the GIC superconductors, which

8
is schematically shown in Fig. 4(a). Actually, exactly
the same model was proposed in as early as 1982 by thepresent author for describing superconductivity in alkali
GICs80 .

FIG. 4: (a)Simplified model to represent M Cx superconductors. We consider the attraction between the 3D electrons in
the interlayer band induced by polar-coupled charge fluctuations of the cation M Z+ and the anion C−δ . (b)Diagrammatic
representation of the equation in the RPA to calculate the effective electron-electron interaction V (p−p′ , iΩ), which will
be substituted in Eq. (6) to evaluate the kernel of the gap
equation.

In order to give some idea about the mechanism to
induce an attraction between 3D electrons in this model,
let us imagine how each conducting 3D electron sees the
charge distribution of the system. First of all, there arepositively charged metallic
M Z+ with its density√ ions2nM , given by nM = 4/3 3 a dx, where a is the bondlength between C atoms on the graphite layer (which is
1.419Å). Note that with use of this nM , the density of
the 3D electrons n is given by (1 − f )ZnM . There arealso negatively charged carbon ions C−δ with the averagecharge of δ ≡ −f Ze/x. Therefore the 3D electrons will
feel a large electric field of the polarization wave comingfrom oscillations of M Z+ and C−δ ions created by either
out-of-phase optic or in-phase acoustic phonons.
We shall consider the coupling of those phonons withthe 3D electrons in terms of the point-charge model, allowing us to write the phonon-exchange polar-coupledinteraction W0 (q, ω) for the scattering of the 3D electrons with momentum- and energy-transfers of q and ωasωp2 (1−f )2
ω 2 −ωLA (q)2ω̄p2 (M̄ /MM +f M̄ /xMC )2,+ V0 (q)ω 2 −ωLO (q)2

W0 (q, ω) =V0 (q)with ωp and ω̄p defined, respectively, assr
4πe2 Z 2 nM
4πe2 Z 2 nM,
and ω̄p =ωp =MM + xMC
M̄

(8)

(9)

where MM and MC are, respectively, the atomic massesof M and C, M̄ (= MM xMC /(MM + xMC )) is the reduced mass of M Cx , ωLO (q) and ωLA (q) are the energies of LO- and LA-phonons, respectively, and V0 (q) is
the bare Coulomb interaction 4πe2 /q 2 . (The subscript 0
indicates that it is the bare interaction to be screened byboth 2D and 3D mobile electrons.)
Owing to the coupling with valence electrons, bothωLO (q) and ωLA (q) depends on f , but the f -dependence
is not important, if we write the phonon-mediated interaction in terms of the corresponding transverse phonon
energies, ωTO (q) and ωTA (q). Thus we specify thephonon energies in terms of ωTO (q) and ωTA (q). In actual calculations, we assume that ωTO (q) = ωt (= constant) and ωTA (q) = ct |q| with ωt of the order of 150K
and ct of the order of 105 cm s−1 for the oscillation perpendicular to the graphite plane.

C.

Calculation of Tc for Alkali-Doped GICs

FIG. 5: Calculated results for Tc as a function of the branching ratio f for alkali GIC superconductors in which Z = 1
and m∗ = me .

By combining this polar-phonon-mediated attractive
interaction W0 (q, ω) with the bare Coulomb interaction
between electrons V0 (q) on the same footing and considering the polarization effects of both 2D and 3D electrons,
we faithfully calculate V (q, ω) the effective interaction
between 3D electrons in the RPA (see, Fig. 4(b)). The
obtained V (q, ω) is put into the kernel, Eq. (6), of thegap equation (4) to obtain Tc from first principles. The
calculated results for Tc in alkali GICs are plotted as afunction of f in Fig. 5 to find that the overall magnitude of Tc is in the range of 0.1 − 0.01K for f ≈ 0.5,
in good agreement with experiment. Note that smaller
values of Tc are obtained for heavier alkali atoms becauseof the smaller couplings as characterized by both ωp andω̄p . This success indicates that the present simple model
applies well at least to alkali GIC superconductors.

9
VII.

SUPERCONDUCTIVITY IN
ALKALINE-EARTH GICS
A.

CaC6

Now let us consider alkaline-earth GIC superconductors. We shall investigate them by adopting the same
simple model with using exactly the same calculation
code developed in 1982 in order to see whether the modeland therefore the piture on the mechanism of superconductivity successfully applied to alkali GIC superconductors can also be relevant to these newly-synthesized superconductors or not28,29 . The parameters specifying themodel will be changed in the following way, if CaC6 isconsidered instead of KC8 :a) Because the valence Z changes from monvalence todivalence, the atractive interaction W0 , which is in
proportion to Z 2 , increases by four times.
b) The interlayer distance d decreases from 5.42Å to
4.524Å, so that the 3D electron density n increases.
c) The factor f to determine the branching ratio decreases from about 0.6 to 0.16.
d) The effective band mass for the 3D interlayer bandm∗ increases from me to about 3me .
e) The atomic number of the ion A hardly changes
from 39.1 to 40.1.

due to the 2D π electrons, which makes the polarcoupled interaction weak, becomes smaller with thedecrease of f .
2) The enhancement of Tc by about one order isbrought about by doubling Z, if m∗ is kept to be
the same value.
3) The enhancement of Tc by about one order is alsobrought about by tripling m∗ from me to 3me , if Z
is taken as Z = 2.
Based on these observations, we can conclude that theenhancement of Tc in CaC6 by about a hundred times
from that in KC8 is brought about by the combined effects of doubling Z and tripling m∗ . In this respect,
the value of m∗ is very important. Appropriateness ofm∗ ≈ 3me is confirmed not only from the band-structure
calculations22,24 but also from the measurement of theelectronic specfic heat20 compared with the corresponding one for KC8 93 .
B.

Other alkaline-earth GICs

Similar calculations are done for other alkaline-earth
GIC superconductors as shown in Fig. 7 in which m∗ isdetermined so as to reproduce EF supplied by the bandstructure calculation. We see that although we give Tc alittle larger than the experimental one for SrC6 , overall
good agreement is obtained between theory and experiment, implying that our simple model may be regarded
as the standard one for describing the mechanism of superconductivity in GICs.

FIG. 6: Calculated Tc as a function of f for m∗ in the rangeof me − 4me with other parameters suitably chosen for CaC6 .
The experimental result is reproduced well, if we choose m∗ ≈3me .

With paying attention to these changes of the parameters, we have calculated Tc for CaC6 as a function of
f . The results are plotted in Fig. 6, from which we canlearn the following points:
1) Overall, Tc becomes higher for smaller f . This canbe understood by the fact that the screening effect

FIG. 7: Calculated Tc as a function of f for alkaline-earth
GICs with m∗ determined so as to reproduce EF provided by
the band-structure calculation.

Here a note will be added to the case of YbC6 ; thebasic parameters such as Z, f , and m∗ for YbC6 areabout the same as those for CaC6 , according to the bandstructure calculation. The only big change can be seen in

10
the atomic mass; Yb (in which A = 173.0) is much heavier than that of Ca by about four times, indicating weakercouplings between electrons and polar phonons as just in
the case of comparison between KC8 and CsC8 . In fact,
Tc for YbC6 becomes about one half of the corresponding result for CaC6 , which agrees well with experiment.
One way to understand this difference is to regard it as
an isotope effect with α ≈ 0.522 .
C.

ings to form the Cooper pairs are restricted, so that Tc
will be suppressed from the value obtained in the standard model. Note that, though its size is much smaller,
this truncation or missing is also seen in SrC6 and thus
the reduction of Tc in experiment for SrC6 from thatpredicted in the standard model may be ascribed to thiseffect.

BaC6The experimental results for Tc in the alkaline-earth
GICs treated in Fig. 7 are also well reproduced by thethe conventional Eliashberg theory in which the McMillan’s formula for Tc is employed with use of the electronphonon coupling constant λ, the average phonon energyω0 , and the Coulomb pseudopotential µ∗ with its conventional value of µ∗ = 0.14. The two parameters, λ andω0 , are determined by the first-principles calculation of
the Eliashberg function α2 F (ω)94 .
This success of the Eliashberg theory is, however, limited; the same theory predicts that BaC6 superconducts
at Tc = 0.23K, but it turns out that superconductivity
does not appear at least down to 80mK95 . In searchof the reason for this discrepancy between theory andexperiment, the phonon structure is extensively studiedin comparison with the case of CaC6 96,97 , but no persuasive reason has been found. In the present author’sview, this failure is directly connected with the problemof obtaining an unrealistically large contribution fromthe intralayer high-energy carbon oscillations to α2 F (ω),
as mentioned in the first paragraph in Sec. V. In fact,
ω0 = 22.44meV is obtained for BaC6 94 , which is muchhigher than the energy of Ba oscillations (≈ 8meV), indicating that the carbon modes are responsible for theunsuccessful prediction of Tc = 0.23K in the conventional
Eliashberg theory.
Very recently BaC6 is discovered to exhibit superconductivity with Tc = 65mK19 . In the framework of thestandard model, m∗ , f , and Z are the important parameters to be determined by the band-structure calculation,from which we see that we may take Z = 2 and f in therange of 0.1−0.3, the same situation as those in CaC6 andSrC6 . As for m∗ , it becomes smaller than 3me , because
the interlayer 3D band of graphite is hybidized with themore itinerant 5d orbitals in BaC6 compared with the 3dones in CaC6 ; if we compare the dispersion relation forthe 3D band along Γχ direction for BaC6 as shown in
Fig. 8(a) with that for CaC6 given in Fig. 3(a), we findthat m∗ ≈ 1.9me . In addition, the 3D band at L point
is located below the Fermi level due to the shorter Brilloiun zone (or equivalently the longer lattice constant)for BaC6 , indicating that some portion of the otherwise
spherical Fermi surface is truncated or missing, as seenin Fig. 8(b) which displays the Fermi surfaces for the 3Dinterlayer bands in CaC6 , SrC6 , and BaC6 . Because ofthis truncation or missing, the virtual multiple scatter-

FIG. 8: (a) Band structure of BaC6 . (b) The Fermi surfaces
for the 3D interlayer bands in CaC6 , SrC6 , and BaC6 94 . (c)Calculated Tc as a function of the optic dielectric constant
ε∞ for BaC6 with m∗ in the range me − 3me . Note that Tc
is 65mK experimentally19 .

This missing of the sperical Fermi surface implies the
reduction of the density of states for the 3D interlayerband at the Fermi level and therefore it might be effectively taken into account by the reduction of m∗from that in the band-structure calculation. Probablym∗ ≈ 1.5me will be a reasonable choice. With this idea
in mind, we have calculated Tc for BaC6 with m∗ in therange me −2me to find that, irrespective of f taken in therange of 0.1 − 0.5, the obtained Tc is always larger than
0.1K, which is about the same as that in the Eliashberg
theory but is much higher than the experimental value.
Thus we need to look for another crucial parameter inthe standard model to explain the experimental value of
Tc .
Basically, the standard model assumes the polarcoupling phonon mechanism of superconductivity inwhich, in general, the effect of the optic dielectric constant ε∞ should be included in the theory and can betreated by changing e2 into e2 /ε∞ in Eqs. (8) and (9).
Physically ε∞ is determined by the magnitude of core polarization of constituent atoms or ions. For light atoms
like carbon, the core polarization is negligibly small andthus we may well take ε∞ as unity. Even for Ca2+ , its
polarizability is about 3.2 in atomic units98 , leading to
ε∞ = 1.07. For heavy atoms, however, it can never be neglected; for Ba, the polarizabilities are, respectively, 124
and 10.5 for Ba+ and Ba2+ , which correspond, respectively, to ε∞ = 3.8 and 1.24. By combining these numbers for ε∞ with the fact that the 3D interlayer band iscompletely occupied in the Γ − L direction, making some
portion of the released 6s electrons actually localize near
the Ba2+ site, leading effectively to the state of Ba(2−δ)+ ,we may assume that the effective value for ε∞ is in the11range of 1.5 − 2.0. Then, as we can see in Fig. 8(c), Tc
obtained in the standard model with m∗ ≈ 1.5me fits
well with the experimental one.
VIII.

PREDICTION OF THE OPTIMUM Tc IN
GICS

As we have seen so far, our standard model could havepredicted Tc = 11.5K for CaC6 in 1982 and it is judgedthat its predictive power is very high. Incidentally, theauthor did not perform the calculation of Tc for CaC6 atthat time, partly because he did not know a possibility
to synthesize such GICs, but mostly because the calculation cost was extremely high in those days; a rough estimate shows that there is acceleration in computers by atleast a millon times in the past three decades. This huge
improvement on computational environments is surely aboost to making such first-principles calculations of Tc as
reviewed in Secs. 3 and 4.

FIG. 9: Prediction of Tc as a function of m∗ for various GICs
in pursuit of optimum Tc . We assume the fractional factor
f = 0.

In any way, encouraged by this success in reproducing
Tc in alkaline-earth GICs, we have explored the optimum
Tc in the whole family of GICs by widely changing various parameters involved in the microscopic Hamiltonian.
Examples of the calculated results of Tc are shown in
Fig. 9(a) and (b), in which f is fixed to zero, the optimum condition to raise Tc , and d is tentatively taken as
4.0Å. From this exploration, we find that the most important parameter to enhance Tc is m∗ . In particular, weneed m∗ larger than at least 2me to obtain Tc over 10K,
irrespective of any choice of other parameters, and Tc is
optimized at m∗ near 15me . The optimized Tc depends
rather strongly on the parameters to control the polarcoupling strength such as Z and the atomic mass A; if
we choose a trivalent light atom such as boron to makeωt large, the optimum Tc is about 100K, but the problem
about the light atoms is that m∗ will never become heavy
due to the absence of either d or f electrons. Therefore
we do not expect that Tc would become much larger than
10K, even if BeC2 or BC2 were synthesized. From thisperspective, it will be much better to intercalate Ti or V,
rather than Be or B. Taking all these points into account,
we suggest synthesizing three-element GICs providing aheavy 3D electron system by the introduction of heavy
atoms into a light-atom polar-crystal environment.

IX.

CONCLUSION

In this chapter, by taking account of the common features elucidated by both the band-structure calculationand the various measurements on the normal-state properties, we have constructed the standard model pertinent
for the description of the mechanism of superconducitity
in metal GICs and then made first-principles calculationsof Tc in the G0 W0 scheme, directly from the microscopic
Hamiltonian representing the standard model. With suitably choosing the parameters in the microscopic Hamiltonian, we have found surprisingly good agreement between
theory and experiment for both alkali and alkaline-earth
GICs, in spite of the fact that Tc varies more than threeorders of magnitude. In this way, we have clarified thatsuperconductivity in metal GICs can be understood by
the picture that the 3D electrons in the interlayer band
supplied by the ionization of metals feel the attractive
interaction induced by the virtual exchange of the polarcoupled phonons of the metal ions. We have also predicted a further enhancement of Tc well beyond 10 K
with giving some suggestions to realize such superconductors in the family of GICs.
By first-principles we usually mean the calculations
based on not the model but the first-principles Hamiltonian. Thus it might be considered as inappropriate tocall the present G0 W0 scheme first-principles, but it is
not an easy task to specify the key parameters to control Tc by just implementing the calculations based on
the first-principles Hamiltonian. We can identify the importance of the parameters, m∗ and Z, only through thecalculations based on the model Hamiltonian, leading to
the better and unambiguous understanding of the mechanism of superconductivity without involved too muchinto the very details of each system which sometimes obscure the essence in first-principles approaches. Besides,
because of the errors involved in the numerical calculations of normal-state properties as mentioned in Sec. 4,
more accurate results of Tc will be obtained by way ofa suitable model Hamiltonian rather than directly fromthe first-principles one.
As a project in the future, it would be important toconstruct a more powerful scheme for the first-principles
calculation of Tc by the combination of the schemes inSecs. 3 and 4, based on which we may make more detailed
suggestions to synthesize GIC superconductors with Tc

12much larger than 10K.

∗

Email: takada@issp.u-tokyo.ac.jp; published in Reference
Module in Materials Science and Materials Engineering(Saleem Hashmi; editor-in-chief), Oxford, Elsevier, 2016;
DOI:10.1016/B978-0-12-803581-8.00774-8
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