Theory for Reliable First-Principles Prediction of the Superconducting Transition
Temperature
Yasutami Takada1, ∗1

arXiv:1601.03486v1 [cond-mat.supr-con] 14 Jan 2016

Institute for Solid State Physics, University of Tokyo,
5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, JapanA review is given for the theoretical framework to give a reliable prediction of the superconducting
transition temperature Tc from first principles, together with a practical strategy for its applicationto actual materials with illustrations of the results of Tc calculated for superconductors in the
weak-coupling region like the alkali and alkaline-earth intercalated graphites as well as those in thestrong-coupling region like the alkali-doped fullerides.
PACS numbers: 74.70.Wz,74.20.-z,74.20.Pq

I.

INTRODUCTION

In quantum mechanics, a ground state is determined through a compromise between the kinetic energy(which makes particles itinerant) and the potential energy (which makes them localized). If the latter includes
the interaction between particles, there appears a further
complication due to their correlated motion. In elucidating the microscopic mechanism of superconductivity,
this intrinsic complexity in quantum mechanics cannot
be avoided but is even more intensified, specifically because superconductivity is a phenomenon in which an
assembly of electrons, negatively charged particles with
one-half spin, goes into the pair-condensed phase as aconsequence of the dominance of some effective attractions between electrons mediated by either phonons, plasmons, spin-fluctuations, or orbital-fluctuations over the
short-range Coulomb repulsions, indicating the necessityof deeply understanding and carefully investigating thephysics of this charge-spin-phonon(-orbital) complex before making a reliable evaluation of the transition temperature Tc of this second-order phase transition. Thus
one would imagine that the task of reliably calculating Tc
must be formidably difficult, but the ultimate goal in thetheoretical study of high-Tc superconductivity should be
to construct a good theoretical framework for an accurate prediction of Tc ; without such a theoretical tool, we
could never conduct a research directly and intimately
touched with the most salient feature of high-Tc materials, namely, the very feature that Tc becomes very highin those materials.
McMillan was the first to provide a rather successful
scheme for predicting Tc in the phonon mechanism of superconductivity, starting from the microscopic electronphonon coupled Hamiltonian. The scheme is known as
the McMillan’s formula1 , which was revised later byAllen and Dynes2–4 . The formulae, both original andrevised, are derived from the Eliashberg theory of superconductivity5 and the task of a microscopic calculationof Tc in this framework is reduced to the evaluation of
the so-called Eliashberg function α2 F (ω) from the firstprinciples Hamiltonian, where F (ω) is the phonon density of states which may be observed by neutron diffrac-tion. This function α2 F (ω) enables us to obtain boththe nondimensional electron-phonon coupling constant λand the average phonon energy ω0 , through which we
can give a first-principles prediction of Tc with an additional introduction of a phenomenological parameterµ∗ (the Coulomb pseudopotential6 ) for the purpose ofroughly estimating the effect of the short-range Coulomb
repulsion between electrons on Tc .
At present, this framework is usually regarded as thestandard one for making a first-principles prediction of Tc
and widely used. In fact, the superconducting mechanismof many (so-called weakly-correlated) superconductors is
believed to be clarified by employing this scheme. The
key phonon modes to bring about superconductivity are
identified by investigating the structure of α2 F (ω). We
can mention that superconductivity in MgB2 with Tc =39K provides a very good example7–10 to illustrate thepower of this scheme. The case of CaC6 with Tc = 11.5K
seems to constitute another recent example11,12 .
In spite of these and many other successful examples,
however, this is not considered to be our ultimate scheme
for calculating Tc from first principles, primarily because
a phenomenological parameter µ∗ is included in the theory. Actually, it cannot be regarded as the method ofpredicting Tc in the true sense of the word, if the parameter µ∗ is determined so as to reproduce the observed Tc .
Besides, as long as µ∗ is employed to avoid a serious investigation of the effects of the Coulomb repulsion on superconductivity, this scheme cannot be applied to stronglycorrelated superconductors such as the high-Tc cuprates.
Even in weakly- or moderately-correlated superconductors, this scheme cannot treat superconductivity originating from the Coulomb repulsion via charge, spin, and/or
orbital fluctuations (namely, the electronic mechanism
including the plasmon mechanism13,14 ). Furthermore, inthis scheme, we cannot investigate the competition or
the coexistence (or even the mutual enhancement due tothe quantum-mechanical constructive interference effect)
between the phonon and the electronic mechanisms.
The validity of the concept of µ∗ is closely related tothat of the Eliashberg theory itself; the theory is valid
only if the Fermi energy of the superconducting electronicsystem, EF , is much larger than ω0 . Note that under the2condition of EF ≫ ω0 , the dynamical response time forthe phonon-mediated attraction ω0−1 is much slower thanthat for the Coulomb repulsion EF−1 , precluding any possible interference effects between two interactions, so thatphysically it is very plausible to separate them. After thisseparation, the Coulomb part (which was not anticipatedto play a positive role in the Cooper-pair formation) has
been simply treated in terms of a single parameter µ∗ .
Thus, for the purpose of searching for some positive roleof the Coulomb repulsion in superconductivity, the concept of µ∗ is irrelevant from the outset of the whole theory.
As for the condition of EF ≫ ω0 , it must also be notedthat such a condition is violated in some recently discovered superconductors in the phonon mechanism including
the alkali-doped fullerenes with Tc = 18−38K15–18 . Once
it is violated, we need to include higher-order correctionsin the electron-phonon coupling (or the so-called vertex
corrections Γ) in calculating the phonon-mediated attractive interaction19 . Then, it is by no means clear whether
we can fully treat the overall effect of various phonons interms of the sum of the contribution from each phonon.
This implies that the Eliashberg function α2 F (ω) will not
be appropriate enough to describe the phonon-mediatedattraction because of possible interference effects among
virtually-excited different phonon modes. As a consequence, λ will not be simply the sum of λi the contribution from the ith phonon, unless λi is small enough tovalidate the whole calculation in lowest-order perturbation.
If the condition of EF ≫ ω0 is violated, especially if EF
is about the same as ω0 , another complication occurs intreating the screening effect of the conduction electrons.
In the usual calculation scheme from first principles, thestatic screening is assumed in calculating α2 F (ω), but it
does not reflect the actual screening process working during the formation of Cooper pairs. This subtle problemof screening is, of course, also closely related to the problem of the first-principles determination of µ∗ and we will
not be able to solve these problems unambiguously without confronting with a difficult task of treating both the
Coulomb repulsion and the phonon-mediated attraction
on the same footing in the calculation of the microscopic
dynamical electron-electron effective interaction V .
In order to overcome the above-mentioned problems inherently associated with the Eliashberg theory, the firstand natural option would be to improve on it by considering both the gap equation and the electron-electron
effective interaction V in entire energy- and momentumspace with properly including the vertex corrections Γ in
V and without separating the Coulomb repulsion fromV . However, this will not be easily accomplished at leastin the near future, partly because the demand for computational resources becomes too much in the solution of
the full nonlocal and dynamical gap equation and partly
because no controlled approximation scheme has been
known for Γ for the superconducting state. (Note thatthe controlled scheme is known for the normal state20,21 .)Fortunately, an alternative option has already been
proposed by the extension of the density functional theory (DFT) to treating a superconducting state22,23 . This
theory provides a formally exact framework for calcualting Tc from first principles by the solution of the gap
equation only in momentum-space, setting aside thecalculation of other physical quantities except for the
one-electron density n(r). Note that the effect of the
Coulomb repulsion is properly included in this formulation without resort to the concept of µ∗ . Therefore
we shall begin with making a very brief review of thisdensity fuctional theory for superconductors (SCDFT)
in Sec. II. We shall point out that the central quantity
in this framework is the pairing interaction K. Then inSec. III, we shall infer a concrete formula for K definedin terms of the Kohn-Sham orbitals in an inhomogeneouselectron gas by reconsidering the gap equation for the homogeneous electron gas in the weak-coupling region withuse of the Green’s-function method. The formula has
not been proposed so far in the literature in SCDFT,
but by its application to superconductivity in the alkaliand alkaline-earth intercalated graphites24,25 , it turns outthat this is indeed a very good approximate functional
form for K. A prediction for the optimum value of Tc by
using this functional form will be given for this class ofmaterials. In Sec. IV, we shall consider a formula for K in
the opposite limit, namely, in the strong-coupling region,
especially for such superconductors with short coherence
lengths as the alkali-doped fullerides26 . By interpolating
the formulae for K in these two limits, we shall propose
a new functional form for K which is supposed to work
well in the whole range of the coupling strength. The
results of Tc obtained by its application to the fullerene
superconductors and related materials will be shown in
the last subsection of this section. Finally in Sec. V weshall conclude this short article, together with discussing
the direction of future research.

II.

DENSITY FUNCTIONAL THEORY FOR
SUPERCONDUCTORS (SCDFT)A.

Hohenberg-Kohn-Sham Theorem

According to the basic theorem in the density functional theory (DFT) due to Hohenberg and Kohn27 , all
the physical quantities of an interacting electron system
are uniquely determined, once its electronic density inthe ground state n(r) is specified. This implies that every quantity including the exchange-correlation energy
Fxc may be considered as a unique functional of n(r).
The ground-state density n(r) itself can be determinedby the calculation of the ground-state electronic density of the corresponding noninteracting reference system that is stipulated in terms of the Kohn-Sham (KS)equation28 . The concept of the noninteracting reference
system is of central importance in the KS algorithm andthe core quantity in the KS equation is the exchange-

3
correlation potential Vxc (r), which is formally defined as
the first-order functional derivative of Fxc [n(r)] with respect to n(r), namely, Vxc (r) = δFxc [n]/δn(r). It must
be noted that Vxc (r) as well as each one-electronic wavefunction at ith level (usually called as “ith KS orbital”)
with its energy eigenvalue εi in the KS equation has nodirect physical relevance; they are merely introduced forthe mathematical convenience so as to obtain the exactn(r) in the real many-electron system by exploitation of
its one-to-one correspondence to the noninteracting reference system.
This basic Hohenberg-Kohn theorem can be applied
not only to the normal ground state but also to the ordered one on the understanding that the order parameter
itself in the ordered state is regarded as a functional ofn(r). In providing some approximate functional form
for Fxc [n] in actual calculations, however, it would be
more convenient to treat the order parameter as an additional independent variable. For example, in considering the system with a collinear magnetic 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 spin-polarized exchangecorrelation energy functional Fxc [nσ ], based on which thespin-dependent exchange-correlation potential is definedto specify the spin-dependent KS equation for determining nσ (r) from first principles.

B.

Gap Equation in SCDFTIn an essentially similar way, in treating superconductivity in the framework of DFT, it would
be better to construct the energy functional with
employing both n(r) and the electron-pair density(or the superconducting order parameter) χ(r, r ′ )(≡hΨ↑ (r)Ψ↓ (r ′ )i) as basic variables22,23 , leading to
the pair-density-dependent exchange-correlation energyfunctional Fxc [n(r), χ(r, r′ )], where Ψσ (r) is the annihilation operator of σ-spin electron field at position r. In accordance with this addition of the order parameter as a fundamental variable to DFT,
not only the exchange-correlation potential Vxc (r) but
also the exchange-correlation pair-potential ∆xc (r, r ′ ) =−δFxc [n, χ]/δχ∗ (r, r ′ ) appear in an extended KS equation. A beautiful point in this density functional theory
for superconductors (SCDFT) is that the extended KS
equation can be written in the form of the Bogoliubovde Gennes equation appearing in the conventional theory
for inhomogeneous superconductors29. Just as is the casewith 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 theexact result for χ(r, r ′ ), containing all the effects of the
Coulomb repulsion including the one usually treated phenomenologically through the concept of µ∗ . As a result,
we can determine the exact Tc by the calculation of thehighest temperature below which a nonzero solution forχ(r, r ′ ) can be found.
In this framework of SCDFT, we can formally write
down the fundamental gap equation to determine Tc exactly as30
X ∆j ′εj ′∆j = −(1)tanh
Kjj ′ ,
′
2ε
2T
j
c
′
j

where ∆j is the gap function associated with jth KS
orbital. In just the same way as its energy eigenvalue εj(which is measured relative to the chemical potential),∆j is not the quantity to be observed experimentally but
just introduced for the mathematical convenience so as
to obtain the exact Tc by solving this BCS-type equation,
Eq. (1). Similarly, the pairing interaction Kjj ′ , definedas the second-order functional derivative of Fxc [n, χ] withrespect to χ∗ and χ, has not any direct physical meaning,
either, although this is a quantity of primary importancein this gap equation or even in the whole framework of
SCDFT.
Three comments are in order: (i) The functional
derivatives of Fxc [n, χ] might not be well defined, as anticipated by remembering the notorious energy-gap problem in semiconductors and insulators31–33 , but as is ordinally the case, we shall assume that Kjj ′ is a well-definedquantity. (ii) In this formal derivation in SCDFT, the dynamical (or ω-dependent) nature in the electron-electron
multiple scatterings does not manifest itself in either the
gap equation or the pairing interaction, in sharp contrast
with the Eliashberg theory. For this reason, many people
cast doubt on whether the physics leading to µ∗ is actually taken into account in SCDFT. However, due to thefact that there is a very good correspondence between
this gap equation and the one in the G0 W0 approximation to the Elishaberg theory, as will be shown in thenext section, we find that it is possible to include thefull dynamical processes in the Cooper-pair formation in
the framework of SCDFT, as long as the form of Kjj ′is properly chosen. (iii) At T = Tc , Kjj ′ is evaluatedat χ = 0. Thus Kjj ′ must be a functional of only thenormal-state electronic density n(r). Note that each KS
orbital, j or j ′ , determined in the normal state may be
regarded as a functional of n(r), justifying the view that
Kjj ′ at T = Tc is eventually a functional of n(r) in thenormal state.
C.

Application and Discussion

This formal framework of SCDFT was not applied toactual superconductors before the year 2005 when an attempt was made to provide a concrete approximate form
for Fxc [n, χ] in which the contribution from the phononmediated attraction was explicitly included up to thelevel of the Eliashberg theory34 . Since then many (butmostly weakly-correlated) superconductors have been analyzed rather successfully in this framework35–40 .
In the judgement of the present author, the presently
available form for Fxc [n, χ] or the one for Kjj ′ contains

4
the information equivalent to that included in the Eliashberg theory for the part of the phonon-mediated attraction, indicating that no vertex corrections are considered in this treatment (amounting to the very insufficient treatment of the strong polaronic effect), while
for the part of the Coulomb repulsion, it contains only
very crude physics; the screening effect is treated in theThomas-Fermi static-screening approximation, or the result in the random phase approximation (RPA) onlyin the static and the long-wavelength limit, neglecting
both the dynamical and nonlocal feature in the effects of
the Coulomb repulsion. This clearly indicates that the
Coulomb repulsion is not treated on the same footing as
the phonon-mediated attraction and this approximation
for the Coulomb part will be just good for describing thephysics represented by µ∗ at usual metallic densities (orrs ≈ 2 with rs the conventional nondimensional density
parameter) from first principles, but it fails to take careof the detailed dynamical nature of the screening effect,
especially, the positive role of the plasmons in superconductivity for lower densities (or larger rs )13,14 . Furthermore, the presently available form for Fxc [n, χ] or Kjj ′does not allow to discuss other types of the electronicmechanisms such as the spin-fluctuation one, either. In
view of these fundamental problems, it is absolutely necessary to derive a much better approximate functional
form for Kjj ′ for the purpose of investigating the electronic mechanisms in the absence/presence of the phononmechanism.
It would be appropriate here to make a rather generalcomment on numerical errors. Currently, calculations of
the normal-state properties are done in either the localdensity approximation (LDA) or the generalized gradientapproximation (GGA)41 to Fxc [n(r)] in DFT. We usually
anticipate that errors in the calculated results are of theorder of 1eV and 0.3eV for LDA and GGA, respectively,
and those errors are much larger than that expected inquantum chemistry (≈ 0.05eV). Now in the usual procedure in SCDFT, the calculation of Tc (which is of theorder of 0.001eV in general) is done simultaneously withthat of the normal state and thus the error for Tc might
be of the same order as that for the normal-state properties, implying that it might become much larger than
Tc itself.
This unfavorable situation may be avoided, if we takethe following two-stage strategy for the calculation of Tc
for a family of superconducting materials in consideration; in the first stage, combined with available experimental results on the normal state, we establish a goodmodel system representing this family of superconductors by making a first-principles band-structure calculation and then in the second stage, we evaluate Tc basedon the model system not only for reproducing the experimental Tc but also for suggesting not yet synthesized but
promising superconductors with higher Tc in this family.
In the rest of this article, we shall discuss two families of
the carbon-based superconductors for which Tc s are calculated and predicted in accordance with this two-stage

strategy.
III. G0 W0 APPROXIMATION WITH
APPLICATION TO GRAPHITE
INTERCALATION COMPOUNDSA.

Pairing Interaction in the Weak-Coupling
RegionThe three-dimensional (3D) homogeneous electron gas
has been known to be a very useful system in constructing a successful functional form for Vxc (r) in either LDA
or GGA by its study with use of various powerful manybody techniques including quantum Monte Carlo simulations. In view of this success, we shall study superconductivity in the same system with the conventional
Green’s-function method in order to infer a good functional form for the pairing interaction Kjj ′ in Eq. (1) thatwill be exact in the weak-coupling limit.
In a homogeneous system, momentum p is always agood quantum number and an electron can be specified
by p and spin σ. If we write the electron annihilation
operator by cpσ , the Hamiltonian H of the 3D electrongas system coupled with phonons is given by
H =He + HphX=ε p c+
pσ cpσ
pσ

+

1 XXX
+
V0 (q)c+pσ cp′ σ′ cp′ −qσ cp+qσ′2
pσ ′ ′q6=0

pσ+ Hph ,

(2)

where εp (≡ p2 /2m∗ −µ) is the bare one-electron dispersion relation with m∗ the band mass and µ the chemical
potential, V0 (q) (≡ 4πe2 /ǫ∞ q 2 ) is the bare Coulomb repulsion with ǫ∞ the optical dielectric constant, and Hphrepresents all contributions containing the phonon operators. For the time being, there is no need of our specifying
a concrete form for Hph .
In the thermal Green’s-function method, we can treat
superconductivity by introducing the abnormal thermal
Green’s function F (p, iωp ), which is defined at temperature T by
Z 1/Tdτ eiωp τ hTτ cp↑ (τ )c−p↓ i.
(3)
F (p, iωp ) = −0

Here ωp is the fermion Matsubara frequency, defined byπT (2p + 1) with an integer p. At T = Tc where thesecond-order superconducting phase transition occurs,
this function satisfies the following formally exact gap
equation:F (p, iωp ) =− G(p, iωp )G(−p, −iωp )XX
˜ p′ ; iωp , iωp′ )F (p′ , iωp′ ), (4)
× Tc
J(p,
ω p ′ p′5where G(p, iωp ) is the normal thermal Green’s function
˜ p′ ; iωp , iωp′ ) is the irreducible electron-electron
and J(p,effective interaction.
Let us assume that the effect of interaction is weak,so that it would be enough to retain the terms only inlowest order in the interaction. If we adopt the same
assumption in the calculation of the normal-state properties in the Green’s-function approach, we are led to theso-called G0 W0 approximation or the one-shot GW approximation in terminology prevailing in the present-day
first-principles calculation community, where W is the effective interaction between electrons including both the
Coulomb and the phonon-mediated interactions and W0
represents W in RPA. Incidentally, in the same kind of
terminology, the Eliashberg theory corresponds to the
GW approximation. Historically, Cohen was the first toevaluate Tc in degenerate semiconductors on the level ofthe G0 W0 approximation42,43 . Unfortunately the pairing
interaction is not correctly derived in his theory, as explicitly pointed out by the present author44 who, instead,by consulting the pertinent work of Kirzhnits et al.45 , has
succeeded in obtaining the correct pairing interaction13 ,the result of which will be reiterated in the following.
In the G0 W0 approximation, we replace G(p, iωp ) by
the bare one G0 (p, iωp ) [≡ (iωp − εp )−1 ] in Eq. (4) and˜ p′ ; iωp , iωp′ ) is well apconsider the case in which J(p,proximated as a function of only the variables, p−p′ andiωp −iωp′ , to write
˜ p′ ; iωp , iωp′ ) = V (p−p′ , iωp −iωp′ ),
J(p,

(5)

as is usually the case for W0 the effective interaction in
RPA, though we do not intend to confine ourselves to
RPA at this stage. By substituting Eq. (5) into Eq. (4),
we obtain the gap equation in the G0 W0 approximationas
F (p, iωp ) = −G0 (p, iωp )G0 (−p, −iωp )XX×Tc
V (p−p′ , iωp −iωp′ )F (p′ , iωp′ ). (6)ω p ′ p′Then, by making an analytic continuation on the ω plane
to transform F (p, iωp ) to the retarded function F R (p, ω)on the real-ω axis and using the general relation due tothe causality principle as
Z ∞
2Ω
dΩ
ImV R (q, Ω), (7)V R (q, ω) = V0 (q)−2 −Ω2 + iηπω0with η a positive infinitesimal, we end up with a gap
equation for F R (p, ω). Finally, by taking the imaginaryparts in both sides of the gap equation and integratingover the ω variable, we are led to an equation depending
only on the momentum variable p. More specifically, theequation can be cast into the following BCS-type gap
equation:∆p = −

X ∆p′
p

2εp′
′

tanh

ε p′
Kp,p′ ,
2Tc

(8)

where the gap function ∆p and the pairing interaction
Kp , p′ are, respectively, defined as
Z ∞dω
Im F R (p, ω),(9)∆p ≡ 2|εp |
π
0
and
Z ∞

ImV R (p − p′ , Ω)2dΩ
Kp , p ≡ V0 (p − p )+Ω+|εp |+|εp′ |
0 π
Z ∞
2
|εp |+|εp′ |=V (p−p′ , iΩ). (10)dΩ 2Ω +(|εp |+|εp′ |)2
0 π
′

′

With use of Kp , p′ thus derived, we can determine Tc
as an eigenvalue of Eq. (8), indicating that we have obtained a scheme in which Tc is given directly from themicroscopic one-electron dispersion relation εp and theeffective electron-electron interaction V (q, iΩ). Because
there is no need to separate the phonon-mediated attraction from the Coulomb repulsion in V (q, iΩ) and the
dynamical nature of the interaction is fully taken into
account, we can properly treat the physics leading to µ∗from first principles with use of this scheme.
The very definition of the gap function ∆p in Eq. (9)indicates that ∆p does not correspond to the physicalenergy gap except in the weak-coupling limit. Similary,
Kp , p′ is not a physical entity, although V is the physicaleffective interaction. Both quantities are introduced forthe mathematical convenience so as to make Tc invariant
in transforming Eq. (4) into Eq. (8). The key point hereis that we need not solve the full gap equation (4) but
much simpler one (8) in order to obtain Tc in Eq. (4).
Of course, if we want to know the physical gap function
rather than ∆p to compare with experiment, we need tosolve the full gap equation, Eq. (4), with Tc determinedby Eq. (8).
Although the spin-singlet pairing has been assumed in
the derivation of Eq. (8), no assumption is made on thedependence of the gap function on angular valuables, sothat this gap equation can treat any kind of the pairing
anisotropy in the gap function, indicating that it can beapplied to s-wave, d-wave, · · · , and even their mixture
like (s+d)-wave superconductors.

FIG. 1: Schematic representation of the dynamical effective
interaction Vjj ′ (iΩ) as inferred from V (q, iΩ).

Now, let us compare Eq. (8) with Eq. (1). Since the KS
orbitals, j and j ′ , can be specified by momenta, p and p′ ,6respectively, in a homogeneous system and εj ′ = εp′ , wemay regard that these two equations are esssentially thesame, suggesting that we may give a concrete functional
form for Kjj ′ with use of energies of the KS orbitals, εjand εj ′ , as
Kjj ′ =Z ∞02
|εj | + |εj ′ |Vjj ′ (iΩ) .
dΩ 2πΩ + (|εj | + |εj ′ |)2(11)

Here Vjj ′ (iΩ) is the dynamical effective interaction working for the scattering process from a pair of electrons∗in (j, j ∗ ) orbitals to another pair in (j ′ , j ′ ) orbitals,
as schematically shown in Fig. 1 (By j ∗ we mean thetime-reversed KS orbital of j.) We should calculate thisVjj ′ (iΩ) on the understanding that it must be derived
from the first-principles Hamiltonian expanded with useof a complete set of the KS orbitals as an orthnormal
basis. Together with the gap equation, Eq. (1), and the
KS orbitals obtained in the normal state by the conventional DFT-based method, Eq. (11) constitutes a basic
framework for a first-principles calculation of Tc for inhomogeneous weak-coupling superconductors.

B.

Superconductivity in Polar Semiconductors

In order to assess the quality of this basic frameworkin the G0 W0 approximation for calculating Tc from firstprinciples, we have applied it to polar degenerate semiconductors, specifically, the doped SrTiO3 and compared
the calculated results with experiments44 .
This material is an insulator and exhibits ferroelectricity under a uniaxial stress of about 0.16GPa along the[100] direction, but it turns into an n-type semiconductor by either 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,
superconductivity 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, implyingthat the superconductivity is brought about by the polarcoupling phonons associated with the stress-induced ferroelectric phase transition,
Taking those situations into account, we have assumed
that the material is well represented by a model of the 3D
electron-gas system coupled with polar-optical phonons
in which a concrete form for V (q, iΩ) can be derived in
RPA asV (q, iΩ) =4πe2,
ǫ(q, iΩ) q 2(12)with the dielectric function in the electron-optical phononsystem as
4πe2
Π0 (q, iΩ)q2
ωt (q)2+ [ǫ0 (q) − ǫ∞ ],ωt (q)2 + Ω2

ǫ(q, iΩ) =ǫ∞ +(13)

where Π0 (q, iΩ) is the polarization function in RPA (orthe Lindhard function) for the 3D electron gas, ωt (q) is
the energy dispersion of the transverse optical phonon,and ǫ0 (q) is the static nonlocal dielectric function, which
is determined with use of the static dielectric constant ǫ0
and ωt (q) as
ǫ0 (q) = ǫ0

ωt (0)2.
ωt (q)2(14)

The dispersionless longitudinal-phonon energy ωℓ is related to the transverse-phonon energy in the longwavelength limit ωt (0) through the Lyddane-Sachs-Teller
relation as
r
ǫ0
ωℓ =
ωt (0).
(15)
ǫ∞
By substituting V (q, iΩ) in Eq. (12) into Eq. (10) andusing the experimental data to determine the values ofparameters like ǫ0 and ǫ∞ as well as the dipersion relation for ωt (q), we have obtained Tc directly from a microscopic model and the results of Tc are in surprizingly
good quantitative agreement with experiment, as shownin Fig. 2 The unusual dependence of Tc on the pressure isalso reproduced well, though it is not shown here. (Forinterested readers, refer to the original paper44 .) This
success indicates that the present framework including
the adoption of RPA for calculating the effective interaction is useful and appropriate at least in the polarcoupled phonon mechanism in which the contribution
from the long-range part of the interaction dominates
over that from the short-range one.
C.

Graphite Intercalation Compounds (GICs)
1.

Historical Survey

The graphite intercalation compounds (GICs) havebeen investigated for a long time from physical, chemical, and technological points of view48–51 . Among various kinds of GICs, special attention has been paid tothe first-stage metal compounds, mainly because superconductivity is observed only in this class of GICs, thechemical formula of which is written as M Cx , where M
represents either an alkali atom (such as Li, K, Rb, andCs) or an alkaline-earth atom (such as Ca, Sr, and Yb)
and x is either 2, 6, or 8. The crystal structure of M Cx isshown in Fig. 3(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.

7FIG. 2: Calculated results for Tc in semiconducting SrTiO3
as a function of the electron density n (the solid curve 44 ),together with the experimental results (the filled circles46 .
See also47 )The first discovery of superconductivity in GICs wasmade in KC8 with the superconducting transition temperature Tc of 0.15K in 196552. 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 1990s53–60 , 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 61 .
A breakthough occurred in 2005 when Tc went up to
11.5K in CaC6 62,63 (and even to 15.4K under pressures
up to 7.5GPa64 ). In other alkaline-earth GICs, the valuesof Tc are 6.5K and 1.65K for YbC6 62 and SrC6 65 , respectively. Since then, very intensive experimental studies
have been made in those and related compounds63,65–67 .
Theoretical studies have also been performed mainly
by making state-of-the-art first-principles calculations of
the electron-phonon coupling constant λ to account forthe observed value of Tc for each individual superconductor11,12,35,68 . Those experimental/theoretical works
have elucidated 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 BCS value69 .
In spite of all those efforts and the existence of sucha generally accepted picture, we need to know more important and fundamental issues that include:(i) Standard model: Can we understand the mechanismof superconductivity in both alkali GICs with Tc in therange 0.01 − 1.0K and alkaline-earth GICs with Tc in
the range 1 − 10K from a unified point of view? In otherwords, is there any standard model for superconductivity
in GICs with Tc ranging over three orders of magnitude?
(ii) Key parameters to control Tc : What is the actual

reason why Tc is enhanced so abruptly (or by about ahundred times) by just substituting K by Ca the atomic
mass of which is almost the same as that of K? In terms of
the standard model, what are the key controlling physicalparameters to bring about this huge enhancement of Tc ?
This change of Tc from KC8 to CaC6 is probably the mostimportant issue in exploring superconductivity across theentire family of GICs.
(iii) Optimum Tc : 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 model andwhat kind of atoms should be intercalated to realize theoptimum Tc in actual GICs?
Recently these three issues have been satisfactorily addressed by the present author24,25 , as shall be explained
one by one in the following three subsections.

FIG. 3: (a) Crystal structure of M Cx (x=2, 6, 8). The case of
x =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) 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 . (c) Superconducting transition temperature Tc in the first-stage alkaliand alkaline-earth-intercalated graphites plotted as a functionof d. The solid circles show the experimental results, while
the crosses the calculated ones in the G0 W0 approximation
for the model represented in (b) with suitable values for theparameters such as Z, m∗ , and f for each M Cx .

82.

Standard Model for Intercalated Graphite
Superconductors

The usual DFT-based self-consistent band-structure
calculation is useful in elucidating the electronic structures of GICs in the normal state, basically because GICs
are not strongly-correlated materials. According to sucha calculation, no essential qualitative difference is foundbetween alkali and alkaline-earth GICs. The main common features among these GICs may be summarized asfollows:
First of all, each intercalant metal atom in M Cx acts
as a donor and changes from a neutral atom M to an
ion M Z+ with valence Z. Then, the valence electrons released from M will transfer either to the graphite π bands
or the three-dimensional (3D) band composed of the intercalant orbitals and the graphite interlayer states70–72 .
We shall define the factor f as the branching ratio between these two kinds of bands. Namely, Zf and Z(1−f )electrons will go to the π and the 3D bands, respectively.
The electrons in the graphite π bands are characterized
by the two-dimensional (2D) motion with a linear dispersion relation (known as a Dirac cone in the case ofgraphene) on the graphite layer.
The dispersion relation of the graphite interlayer bandis very similar to that of the 3D free-electron gas, folded
into the Brillouin zone of the graphite68 . Thus its energy
level is very high above the Fermi level in the graphite,
because the amplitude of the wavefunction for this bandis 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 theenergy level of the interlayer band below the Fermi level.
The dispersion of the interlayer band is modified fromthat of the free-electron gas because of the hybridization
with the orbitals associated with M , but generally it iswell 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 with s-orbitals,
allowing us to consider that m∗ = me , while in alkalineearth GICs, the hybridization with d-orbitals conrtibutes
much, leading to m∗ ≈ 3me in both CaC6 and YbC6 , asrevealed by the band-structure calculation11,12 .
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 0.673,74 . On the other hand, f
is about 0.16 in CaC6 , making the electron density inthe 3D band n increase very much12 . This increase in n
is easily understood by the fact that the energy level ofthe interlayer band is much lower with Ca2+ than with
K+ . The concrete numbers for n are 3.5 × 1021cm−3 and2.4 × 1022cm−3 for KC8 and CaC6 , respectively, in which
the difference in both d and x is also taking into account.
As inferred from experiments49,68 and also from thecomparison of Tc calculated for each band75 , it has beenconcluded that only the 3D interlayer band is responsible

for superconductivity. Note that 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.
With the above-mentioned common features in mind,
we can think of a simple model of a 3D electron gas coupled with phonons for the GIC superconductors, whichis schematically shown in Fig. 3(b). In order to givesome 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 are posivelyZ+charged metallic
with its density nM , given by√ 2 ions M
nM = 4/3 3 a dx, where a is the bond length between C
atoms on the graphite layer (which is 1.419Å). Note thatwith use of this nM , the density of the 3D electrons n isgiven by (1 − f )ZnM . There are also negatively charged
carbon ions C−δ with δ given by δ = f Z/x on the average. Therefore the 3D electrons will feel a large electricfield of the polarization wave coming from oscillations of
M Z+ and C−δ ions created by either out-of-phase opticalor in-phase acoustic phonons.
Although there are some additional complications originating from the combined contributions from both optical and acoustic modes in the layered-lattice system,
this coupling of an electron with the polar phonons isessentially similar to the one we have already consideredin the previous subsection. Thus it is straightforward
to derive the effective interaction V (q, iΩ) in RPA in
which the bare Coulomb repulsion and the polar-phononmediated attraction are treated on the same footing withthe screening effects of both the 2D and 3D electrons. A
concrete form for V (q, iΩ) will not be given here, but for
its detailed derivation we refer to the original paper75 inwhich exactly the same model as presented in Fig. 3(b)was proposed in as early as 1982 by the present author
for analyzing superconductivity in alkali GICs.

3.

Key Physical Parameters to Control Superconductivity

We have evaluated Tc from first principles by using
V (q, iΩ) thus obtained to solve the gap equation (8). In
Fig. 3(c), the calculated results of Tc for various M Cx
are plotted by the crosses with the choice of suitable values for the parameters such as Z, m∗ , and f for eachmaterial. As we see, the agreement between theory andexperiment is quite excellent across the entire family ofGICs, implying that our simple model may well be regarded as the standard one for describing the mechanismof superconductivity in GICs.
In order to identify the controlling physical parameters to enhance Tc in CaC6 by a handred times fromthat in KC8 , let us compare the values for the physicalparameters between the two materials: (i) The valence
Z; because the valence changes from monvalence to divalence, the value of Z in CaC6 is doubled to make the9bare polar-phonon-mediated attraction (which is in proportion to Z 2 ) stronger by four times. (ii) The interlayer
distance d; it decreases from 5.42Å to 4.524Å, so that the3D electron density n increases in CaC6 . (iii) The factor f to determine the branching ratio; it decreases fromabout 0.6 to 0.16, which also makes a further increase
in n. (iv) The effective band mass for the 3D interlayerband m∗ ; it increases from me to about 3me , leading toa large enhancement of the density of states at the Fermi
level. (v) The atomic number of the ion A; it changes
only from 39.1 in K to 40.1 in Ca. Thus the energies ofphonons hardly change.
We have recalculated Tc by shifting each parameter,
one by one, from the above-mentioned respective physical value and have found that two parameters, namely,
Z and m∗ , are very important in controlling the overallmagnitude of Tc . In fact, Tc is enhanced by one order ofmagnitude from that in KC8 by doubling Z from Z = 1
to Z = 2 with m∗ kept to be me . A further enhancementof Tc by another one order is seen by tripling m∗ fromme to 3me , with Z kept to be Z = 2. Thus we may
conclude that the enhancement of Tc in CaC6 by about ahundred times from that in KC8 is brought about by the
combined effects of doubling Z and tripling m∗ . In thisrespect, the actual value of m∗ is very important. Appropriateness of m∗ ≈ 3me is confirmed not only from theband-structure calculations11,12 but also from the measurement of the electronic specfic heat66 compared withthe corresponding one for KC8 76 .
A note will be added here on 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 difference can be seenin the atomic mass; Yb (in which A = 173.0) is muchheavier than that of Ca by about four times, indicating
weaker couplings between electrons and polar phonons asjust in the case of comparison between KC8 and RbC8
or CsC8 . In fact, Tc for YbC6 becomes about one halfof the corresponding result for CaC6 , which agrees wellwith experiment. One way to understand this difference
is to regard it as an isotope effect with α ≈ 0.511 .
4.

FIG. 4: Prediction of Tc as a function of m∗ for various GICs
in pursuit of optimum Tc . We assume the fractional factor
f = 0.

of GICs by widely changing various parameters involvedin the microscopic Hamiltonian of the standard model.
Examples of the calculated results of Tc are shown in
Fig. 4, 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 parameterto enhance Tc is m∗ . In particular, we need m∗ larger
than at least 2me to obtain Tc over 10K, irrespective of
any choice of other parameters, and Tc is optimized form∗ in the range (10 − 20)me . The optimized Tc depends
rather strongly on the parameters to control the polarcoupling strength such as Z and the atomic mass A; if wechoose a trivalent light atom such as boron to make ωt (0)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.

Prediction of Optimum TcIV.

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 a first-principles calculation of Tc
in the G0 W0 approximation.
We have explored the optimum Tc in the whole family

STRONG-COUPLING APPROACH WITH
APPLICATION TO FULLERIDESA.

Coherence Length

In the BCS theory for superconductivity which is applicable to superconductors in the weak-coupling region, Tc
is directly related to the coherence length ξ0 which characterizes the spatial extent of the wave function representing the bound state of a Cooper pair at zero temperature. More concretely, the relation between them is expressed as Tc /EF = 2eγ /π 2 (pF ξ0 )−1 ≈ 0.361/pF ξ0 withγ (= 0.57721 · · · ) the Euler-Mascheroni constant and pF
the Fermi wave number. Because p−1
F is in proportion to10
the lattice constant a0 , the relation is rewritten into theform asTc
a0≈ 0.0735
EF
ξ0

(16)

for a monovalent metal in the fcc-lattice structure. (Forother valence and/or crystal structure, the coefficient of
0.0735 changes, but it always remains in the same order of magnitude.) For usual elemental superconductors,
Tc /EF is of the order of 10−4 and thus ξ0 is about athousand times larger than a0 , validating the approachin momentum space. On the other hand, the relationin Eq. (16) implies that high Tc is inevitably associatedwith short ξ0 . In fact, ξ0 is observed as only a few nm
or less, i.e., of the same order of a0 in many of the recently synthesized high-Tc superconductors77 such as thecuprates, the alkali-doped C60 , and MgB2 .
A similar message can be obtained through the socalled Uemura plot78–80 , according to which there is auniversal relation of Tc /EF ≈ 0.04 for a wide variety ofstrong-coupling superconductors. If this relation is put
into Eq. (16), we obtain ξ0 ≈ 2 a0 . Furthermore, if wethink of the Bose-Einstein condensation (BEC) for an assembly of very tightly-bound pairs of electrons, the condensation temperature (which amounts to Tc in such asystem) is given by Tc /EF = (2/9π ζ(3/2)2 )1/3 ≈ 0.218,
where ζ(3/2) (= 2.6124 · · · ) is the Riemann’s zeta function ζ(x) at x = 3/2. This value of Tc /EF (which must be
the optimum value for fermionic superconductors) suggests ξ0 ≈ 0.3 a0 . Thus, in treating superconductivity inthe strong-coupling region, we need to consider a situation of extremely short ξ0 , validating the approach in real
space, which is totally different from that of very long ξ0
in the weak-coupling superconductors.
B.

Pairing Interaction in the Strong-Coupling
RegionIn view of the above-mentioned difference in ξ0 , weshall exploit the shortness of ξ0 in reformulating the
problem of making a quantitative calculation of Tc forstrong-coupling superconductors in the Green’s-function
approach26. Let us start with this reformulation by considering the dynamical correlation function for singlet
pairing of two electrons in the normal state QR
sc (q, ω),which is defined asZ ∞dt eiωt−ηt h[eiHt Φq e−iHt , Φ+
QR
(q,
ω)
=
−i
q ]i, (17)sc
0where H is the Hamiltonian for a homogeneous electron
system and Φq is P
the electron-pair annihilation operator,
defined by Φq ≡ p cp+q↑ c−p↓ . In terms of the retarded
pairing correlation function QR
sc (q, ω), we can define Tc
as the temperature at which QR
sc (q, ω) diverges at q = 0
in the static limit (ω → 0) with the decrease of T .
As shown schematically in Fig. 5(a), Qsc is conventionally divided into a sum of terms classified by the number

˜ where J,
˜ which has already appeared in Eq. (4), isof Js,
the irreducible electron-electron effective interaction including all vertex corrections. Using Πsc (q, ω) the pairing polarization function composed of two full electron
Green’s functions including all self-energy corrections, we
can express the infinite sum in Fig. 5(a) in a quite symbolical way as
Qsc = −

Πsc.
1 + J˜ Πsc(18)Then, the divergence in QR
sc (0, ω → 0) occurs at the zeroof the denominator in Eq. (18), which provides exactly
the same Tc as that obtained through the solution of
Eq. (4).

FIG. 5: Pair correlation function Qsc described in terms ofusing either (a) the pairing polarization function Πsc and theirreducible electron-electron effective interaction J˜ or (b) thenoninteracting pairing polarization function Πsc,0 and the effective interaction g̃. By the extension of the idea of expanding Qsc with use of Πsc,0 , we show a scheme to express
it in terms of Kohn-Sham orbitals in (c), together with theschematic representation of g̃jj ′ in (d).

Now, instead of using Πsc , we shall consider an alternative expansion with use of Πsc,0 the pairing polarization
function composed of two bare electron Green’s functions.
As shown schematically in Fig. 5(b), if we introduce thepairing interaction g̃ by the definition of11g̃ ≡ J˜ +
−
,
Πsc
Πsc,0

(19)

we can rewrite Qsc in Eq. (18) into another exact form
as
Qsc = −

Πsc,0.
1 + g̃ Πsc,0(20)

Since we can always calculate Πsc,0 easily, the problem of
estimating Tc is reduced to the evaluation of g̃ in the limit11of ω → 0 at q = 0. Note that g̃ does not depend strongly
on either q or ω, in sharp contrast with ΠR
sc,0 (q, ω). In
the BCS theory, for example, g̃ is taken as a constantwhich represents a weakly attractive and spatially local
interaction working only in the vicinity of the Fermi level,
given that the phonon energy ω0 is much smaller than
EF . Then the zero of the denominator in Eq. (20) at
q = 0 and ω → 0 provides the well-known BCS formula
for Tc . Note also that the problem of finding the zero of
1+g̃ ΠR
sc,0 (0, ω → 0) is just the same as that of solving the
gap equation in Eq. (6) with replacing V (p−p′ , iωp−iωp′ )by g̃.
In the case of short ξ0 , we may assume that the essential physics of electron pairing can be captured evenif we treat only a small-cluster system, as long as thesystem size is large enough in comparison with ξ0 . Under this assumption, we can take the following procedurefor determining g̃: First, by representing the values for(N )Πsc,0 and g̃ in an N -site system as Πsc,0 and g̃N , respectively, we may write the pairing correlation function for(N )the N -site system Qsc as
(N )

)
Q(N
sc = −

Πsc,0(N )1 + g̃N Πsc,0

,

(21)

in accordance with Eq. (20). In general, the exact bulk
value g̃ will be obtained by taking the large-N limit ofg̃N , but expecting that g̃N0 with a small positive integer
N0 (for example, N0 ≈ 2) is already close to g̃, we may
estimate g̃ by the relation ofg̃ = −

1(N )Qsc,00−1(N )Πsc,00

,

(22)

(N )

with Qsc 0 evaluated rigorously, for example, by exact
diagonalization. Of course, we can determine a better
value of g̃ by checking the saturation behavior of {g̃N0 }with the increase of N0 .
C. Interpolation Formula for the
Pairing-Interaction Functional

Based on the knowledge so far obtained, let us speculate an appropriate functional form for the pairing interaction Kjj ′ in SCDFT. It would be natural to expand
the electron field operator in terms of the Kohn-Sham orbitals {|ji} in an inhomogeneous electron system, so that
we can define the electron-pair operator Φj as Φj ≡ cj cj ∗with |j ∗ i the time-reversed orbital of |ji. Then, we canintroduce the pairing polarization function in terms of Φjand Φ+
j ′ . In particular, we can calculate this quantity inthe noninteractiong system Πsc,jj ′ in analogy with Πsc,0 ,as schematically introduced in Fig. 5(c). At the sametime, we can define a quantity g̃jj ′ as schematically shownin Fig. 5(d).
In the strong-coupling region, we do not expect that
this g̃jj ′ exhibits strong dependence on iΩ, because theFermi energy, the scale for the electron kinetic energy,
must be much smaller than the energy scale representing
electron-electron interactions. In addition, g̃jj ′ does notdepend strongly on orbital variables, either, because thepairing interaction should be short-ranged in real spacein line with the shortness of ξ0 . Thus we may regardg̃jj ′ as a constant. Then, if we take Kjj ′ in Eq. (1) asg̃jj ′ , we can easily see that both Eq. (1) and the zero of
1 + g̃jj ′ ΠRsc,jj ′ provide the same Tc .
In a more general situation of the intermediatecoupling region, however, g̃jj ′ will depend on both iΩand orbital variables. In order to treat such a situation
and also by paying attention to the weak-coupling situation examined in the previous section, we propose thefunctional form for Kjj ′ in exactly the same form as thatin Eq. (11) with replacing Vjj ′ (iΩ) by g̃jj ′ (iΩ). Notethat this functional form of Kjj ′ is reduced to g̃jj ′ (0),if g̃jj ′ (iΩ) does not depend on Ω, assuring us that theframework in SCDFT with this choice of Kjj ′ provides
the same Tc as that in the usual Green’s-function approach in the strong-coupling region. This means that
we can successfully take care of both self-energy and vertex corrections beyond the G0 W0 approximation by upgrading Vjj ′ (iΩ) to g̃jj ′ (iΩ) in Eq. (11).

D.

Alkali-Doped Fullerides

1.

Aims of This Subsection

The fulleride is a molecular crystal with narrow threefold conduction bands (with the band width W ≈0.5eV) derived from the t1u electronic levels of each
C60 molecule. With the doping of three alkali atoms
per one C60 molecule, we obtain the half-filled situation in which superconductivity occurs with Tc in therange 18 − 38K15,16 and the short coherence length ξ0 of
only a few molecular units. As for the mechanism of
superconductivity in the alkali-doped fullerides, phonons
are widely believed to play an important role. This belief is based upon a crude estimate of Tc in the conventional Eliashberg theory. In fact, analysis of various experiments with use of this theory has shown that many
aspects of superconductivity in these fullerides are consistent with a picture of s-wave BCS superconductors
with the Cooper pairs driven by the coupling to the intramolecular high-frequency phonons in Hg symmetry(with the phonon energy ω0 ≈ 0.2eV and the nondimensional electron-phonon coupling constant λ ≈ 0.6 for
Rb3 C60 )17,81,82 .
The above picture, however, seems to be too much
simplified and we may raise several fundamental questions. From a theoretical point of view, one of the serious
problems is ill foundation to adopt the Eliashberg theory in the fullerides due to the importance of vertex corrections19,83 . From an experimental side, the following
four experimental facts have been observed which we cannot easily understand with use of the Eliashberg theory:

12(i) The relation between Tc and the lattice constant a0changes remarkably when the crystal structure changes
from the face centered cubic (fcc) to simple cubic (sc)lattice by the introduction of Na, a smaller ion compared
to K, Rb, or Cs, as a dopant ion84 . (ii) The antiferromagnetic (AF) insulating behavior has been reported inammoniated K3 C60 , which is peculiar in the sense thats-wave BCS superconductivity exists in the vicinity ofan AF phase85 . A similar problem is also seen in thebody-centered cubic A15-structured Cs3 C60 16 . (iii) The
anomalous 13 C isotope effect on Tc for 50% 13 C substitution has been observed, reflecting the difference between the molecular and atomic mixture of 12 C and 13 C
atoms86 . (iv) With the deviation of electron number persite n from half-filling, Tc decreases rapidly in both sidesof the deviation87 .
Although the experiment (i) may be reproduced in the
Eliashberg theory with some judicious choice of parameters, the rest (ii)-(iv) cannot be explained even qualitatively in the theory. In addition, the behavior of Tc asa function of n observed in the experiment (iv) cannot
be predicted either by those theories proposed so far toaccount for the copper-oxide high-Tc superconductivity
based on some strong-correlation models, though suchmodels may be favorable for the explanation of the experiment (ii).
A successful theory for the fullerene superconductors
should not only reproduce these experiments in a coherent fashion but also clarify the reason why the effect ofvertex corrections, even if it is large, does not manifest itself in many superconducting properties. In quest of sucha theory from a somewhat general viewpoint, the present
author made intensive studies of the Hubbard-Holstein(HH) model and its extension in the past and successfully explained all the issues (i)-(iv) raised above18,88–92 .
In the rest of this subsection, we shall focus on the first
three issues in order to show how nicely the HH model
applies to the fullerides, as far as Tc is concerned. Then
we shall explore a possibility to enhance Tc in this classof superconductors by changing the parameters involvedin the model.

2.

The Hubbard-Holstein Model

In a molecular crystal, it is a very good approximation to regard each molecular unit as a “site” in a lattice. We shall adopt this approximation and describe theelectron-phonon system in the alkali-doped fullerides bya model Hamiltonian H in site representation in which
each site corresponds to each C60 molecular unit. Since
the conduction band width is narrow, the intermolecular
hopping integral t must be small, indicating that only
the nearest-neighbor hopping is relevant in modelling the
kinetic energy of the system. Then, it would be appropriate to decompose H into the nearest-neighborP
electrontransfer term Ht and the sum of site terms
i Hi including both the electron-electron and the electron-local

phonon interactions. In order to faithfully represent thethreefold degenerate t1u -conduction bands coupled witheight Hg intramolecular Jahn-Teller (JT) phonons, wewould need to include the t1u ⊗ Hg JT structure in Hi , as
has often been the case93–96 . In making a detailed study
on the stability of the AF insulating phase, it is known
that this feature of band multiplicity plays a rather crucial role97 , but in treating superconductivity itself, theband multiplicity is not considered to be of primary importance98 . Therefore we shall take a simplest possiblemodel, namely, the one-band HH model to discuss superconductivity from a more general viewpoint that will
be applicable to the whole family of fullerene superconductors including not only electron-doped but also holedoped materials with different JT structures.
The concrete form for H in the HH model is given by
X
X †
Hi ,
(23)
(ciσ ci′ σ + h.c.) +H = −tihi,i′ i σwith the site term Hi , written as
X
Hi = −µniσ + U ni↑ ni↓
σ

+

X
√
αω0
niσ (ai + a†i ) + ω0 a†i ai ,

(24)

σ

where hi, i′ i represents the nearest-neighbor-site pair, ciσ
the operator to annihilate a spin-σ electron at site i,µ the chemical potential, niσ ≡ c†iσ ciσ , U the on-site(or intramolecular) Coulomb repulsion, α the nondimensional electron-phonon coupling constant which is related
to the conventional electron-phonon coupling constant λthrough λ = αω0 /tz with ω0 the optical-phonon energy
and z the coordination number, and ai the operator toannihilate an optical phonon at site i.
The characteristic features of the fullerenes can be well
captured by an appropriate choice of the parameters involved in this Hamiltonian. In fact, the narrowness ofthe conduction band can be described by the smallnessof t of the order of 0.1eV. The difference in the crystal
structure as well as the effect of band multiplicity canbe well accounted for by a suitable choice of z. The
high-frequency intramolecular optical phonons coupledstrongly to electrons can be treated by considering the
local phonons with the energy ω0 (≈ 0.2eV) at each site.
The short-range Coulomb potential must be relevant in
the fullerenes due to the proximity to the AF state and it
can be included by the introduction of U . Note that thisU is not the direct Coulomb repulsion between electrons
on a carbon atom Uatom which is of the order of 5-10eV.
Rather it is the sum of Uatom and the attraction −Upoldue to the electronic polarization effect of 60 π-electrons
in the C60 molecule99 . Because of a strong cancellation
between Uatom and −Upol , U is expected to be of the order of 0.1eV, which is about the same magnitude as thatfor the phonon-mediated attraction, −Uph ≡ −2αω0 .
Intensive studies on the ground state in this system byexact diagonalization of small-size clusters have revealed13
that the half-filled HH model exhibits interesting competition among charge-density-wave (CDW), spin-densitywave (SDW), and superconducting states89,91 . We may
summarize the results in the following way: (a) If U −Uphis at least smaller than −t, the CDW state composedof an array of immobile bipolarons is stabilized. (b) If
U − Uph is larger than t, there appears the SDW state
which is nothing but the AF state in this case. (c) Superconductivity appears only in the CDW-SDW transition
region where |U − Uph | is less than t. In this offset situation of U ≈ Uph , the state may be regarded as an
assembly of nearly free polarons and the main effect of
the strong electron-phonon vertex corrections is found toform a polaron from a bare electron. This implies that
the Eliashberg theory or even the BCS theory is expected
to be accurate enough to describe superconductivity inthis system, if it is applied on the basis of the polaron
picture89,91,100 . In any case, the intrisic competition ofsuperconductivity with the AF state in the half-filled HHmodel resolves the issue (ii) raised previously.

√with d = a0 / 2 − 6.95Å , ∆ = 0.55Å , and the corresponding parameters, tRb and dRb , for Rb3 C60 . By
choosing tRb = 1.70ω0 suitable for Rb3 C60 and thecommon parameters such as U = 3.81ω0, α = 2, andω0 = 0.195eV with reference to band-structure calculations and relevant experiments, we obtain the results for
Tc as a function of a0 , which agrees remarkably well withexperiment as clearly demonstrated in Fig. 6 Note thatin our calculation, the difference in Tc between fcc andsc structures arises mainly from that in the “effective”
lattice coordination number z 82 . This indicates that z isa key to the resolution of the issue (i). Note also that the
recent experimental result for Cs3 C60 under pressure isalso on our theoretical curve, if we estimate an appropriate value of a0 so as to reproduce the same volume per
one C60 molecule for this bcc A15 structure.
The competing feature between Uph and U is also identified to be a key to the resolution of the issue (iii)90 . The
observed anomalous isotope effect cannot be explained byeither the phonon or the electronic polarization mechanism alone. It can be reproduced, if both these mechanisms are included simultaneously. Namely, we need toconsider the change in both ω0 and U induced by theisotope substitution. The sensitivity of Tc to the local
change in the Coulomb potential U is due to the very
short-range nature of ξ0 of this superconductivity.
3.

FIG. 6: Tc as a function of the lattice constant a0 in the alkalidoped fullerides and its dependence on the crystal structure.
The solid and the broken curves represent our theoretical results, while the experimental results are given by square, triangle, diamond, and circle marks.

In order to address the issue (i), we have applied ourframework to calculate Tc to the HH model by estimating g̃ with use of a cluster as small as two molecules (i.e.,
N0 = 2). We have evaluated the variation of t with thechange of the lattice constant a0 by fitting the change inthe band width obtained by the band-structure calculation for each lattice structure101,102 . More specifically, t
is determined bydd − dRbt = tRb
,
(25)
exp −
dRb
∆

Prospect for Room-Temperature Superconductors

Encouraged by the success in reproducing Tc for thealkali-doped fullerene superconductors by the application of our theoretical framework to the HH model at
half-filling in the nearly offset situation of U ≈ Uph , wehave attempted to explore the optimum value of Tc with
the increase of the electron-phonon coupling constant α,
whereby U is also increased to keep the offset situation.
Examples of the calculated results of Tc are plotted inFig. 7 As is seen, Tc goes beyond 100K for α = 3 and it
reaches room temperature for α = 4.
By the band-structure calculation103 , it is known that
α is about three for the hole-doped C60 in which holecarriers will be introduced into the hu -valence bands. On
the other hand, in the crystal composed of C36 , α will be
about four104 , if the C36 solid is successfully doped tohave an enough number of mobile carriers in the system,
although it seems to be a very difficult task105 .

V.

CONCLUSION AND DISCUSSION

In this chapter, we have proposed a new functional
form for the pairing interaction Kjj ′ , a key quantity inthe gap equation, Eq. (1), to determine Tc in the density functional theory for superconductors. The functional form is given in Eq. (11) with the effective electronelectron interaction Vjj ′ (iΩ) replaced by g̃jj ′ (iΩ) defined
schematically in Fig. 5(d). We have assessed the use-

14

FIG. 7: Tc as a function of the lattice constant a0 . For
electron-doped C60 with α = 2, the theoretical curves are
drawn, together with the experimental results. We also givethe results for both α = 3 and 4 corresponding to (hypothetical) hole-doped C60 and doped C36 .

both the weak-coupling superconductors like the alkaliand alkaline-earth-intercalated graphites and the strongcoupling superconductors like the alkali-doped fullerides.
We have also explored a possibility to enhance Tc up to
room temperature.
The proposed functional form has just opened a challenging frontier of the research on high-Tc superconductivity. Although we need to improve on the functional
form itself by utilizing the information obtained by its
application to a much wider range of materials, we canthink of several ways to make use of this new theoretical
tool. For example, we can make a quantitative assessment of the effectiveness of each superconducting mechanism, either phononic or electronic, so far proposed byinvestigating how much Tc is actually enhanced with theintroduction of the mechanism. The knowledge accumulated by such investigations will pave the way to reach aroom-temperature superconductor.

VI.

ACKNOWLEDGMENTS

fulness of this functional form by its applications toThis work is supported by a Grant-in-Aid for Scientific
Research (C) (No. 21540353) from MEXT, Japan.

Email: takada@issp.u-tokyo.ac.jp; published in “Carbonbased superconductors: Toward high-Tc superconductivity”, edited by J. Haruyama (Pan Stanford, Singapore,2015), pp. 193-230; ISBN: 978-981-4303-30-9 (Hardcover), 978-981-4303-31-6 (eBook).
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