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arXiv:cond-mat/0105351v2 [cond-mat.supr-con] 18 Feb 2002
High Tcsuperconductivity in MgB2by nonadiabatic pairing
E. Cappelluti1, S. Ciuchi2, C. Grimaldi3, L. Pietronero1,4, and S. Str¨assler3
1Dipart. di Fisica, Universit´a di Roma “La Sapienza”, Piazzale A. Moro, 2, 00185 Roma, and INFM UdR Roma1, Italy
2Dipart. di Fisica, Universit´a dell’Aquila, v. Vetoio, 67010 Coppito-L’Aquila, and INFM, UdR l’Aquila, Italy
3Ecole Polytechnique F´ed´erale de Lausanne, IPR-LPM, CH-1015 Lausanne, Switzerland
4Istituto di Acustica “O.M. Corbino”, CNR, Area di Ricerca Tor Vergata, Roma, Italy
The evidence for the key role of the σbands in the electronic properties of MgB2points to the
possibility of nonadiabatic effects in the superconductivity of these materials. These are governed
by the small value of the Fermi energy due to the vicinity of the hole doping level to the top of the
σbands. We show that the nonadiabatic theory leads to a coherent interpretation of Tc= 39 K
and the boron isotope coefficient αB= 0.30 without invoking very large couplings and it naturally
explains the role of the disorder on Tc. It also leads to various specific predictions for the properties
of MgB2and for the material optimization of these type of compounds.
The field of high-Tcsuperconductivity is living an ex-
citing time [1]. New techniques provide in fact the
possibility to explore physical regimes that were previ-
ously inaccessible and superconducting materials which
were often regarded as “conventional” BCS ones, as the
fullerenes, have proven to be real high-Tccompounds [2].
In this context the magnesium diboride MgB2, which was
recently found to be superconductor with Tc= 39 K [3],
is a promising material. The question is to assess whether
MgB2is one of the best optimized BCS materials or its
superconducting properties stem from a novel mechanism
of pairing and can be further improved in MgB2or in re-
lated compounds. In this Letter we would like to discuss
some theoretical and experimental evidences that in our
opinion point towards an unconventional type for the su-
perconductivity, which we identify with the nonadiabatic
framework.
MgB2is often regarded in literature as a conventional
BCS-like superconductors, whose properties could be well
described by the standard Migdal-Eliashberg (ME) the-
ory. The high value of Tcis thus ascribed to the high
frequency B-B phonon modes in the presence of a in-
termediate or strong electron-phonon (el-ph) coupling λ.
LDA calculations find in fact λ≃0.7−0.9 which, all
together with the a representative phononic energy scale
ωph ≃650 −850 K, is in principle able to account for the
large value of Tcin MgB2[4–8]. However this picture is
shaken by a series of facts. First, recent reflectance data
are not consistent with a value of λstrong enough to give
Tc= 39 K [9,10]. Second, the experimental determina-
tion of the total isotope effect on Tcreported a boron
isotope coefficient αB= 0.30 and a negligible magnesium
isotope effect [11]. Preliminary indications suggest that
this value of αcannot be explained by the LDA estimates
of λ≃0.7−0.9, but requires a much larger coupling
λ≃1.4 [11]. We have solved numerically the Eliashberg
equations to reproduce the experimental value of the iso-
tope coefficient. We consider a rectangular Eliashberg
function [α2F(ω) = const. for 650 K ≤ω≤850 K] as
well as a simple Einstein spectrum with frequency ω0.
1.0 1.5 2.0 2.5 3.0
λ
0
20
40
60
80
Tc [K]
1.0 1.5 2.0 2.5 3.0
λ
0
20
40
60
80
Tc [K]
1.0 1.5 2.0 2.5 3.0
λ
0
20
40
60
80
Tc [K]
FIG. 1. Critical temperature Tcas function of λfor fixed
value of boron isotope effect αB= 0.30. Solid line corresponds
to the rectangular Eliashberg function, grey region represents
the solutions spanned by the Einstein model with frequency
650 K ≤ω0≤850 K. The dashed line marks the value Tc= 39
K.
The limiting values ω0= 650 K and ω0= 850 K of the
Einstein model can be thus considered respectively as
lower and upper bounds of a realistic Eliashberg func-
tion. In Fig. 1 we show the critical temperature Tcas
function of λfor fixed value of α= 0.30. The correspond-
ing needed Coulomb pseudopotential varies in the range
µ∗≃0.28 −0.30 and does not depend on the specific
Eliashberg function. We can see that a quite strong el-
ph coupling is required to reproduce both Tc= 39 K and
α= 0.30 with λranging from 1.4 to 1.7. These values of
λand µ∗are thus even larger than the estimations of Ref.
[11], confirming and reinforcing the discrepancy between
LDA results and the ME analysis of the experimental
data (see also Ref. [12]). Note that, contrary to cuprates
and fullerenes, electronic correlation is not expected to
play a significant role in MgB2, and LDA calculations
should be considered quantitatively reliable.
This analysis therefore points towards a more complex
framework to understand superconductivity in MgB2.
An useful insight, in our opinion, comes from a com-
parison of the electronic structure of MgB2and graphite.
1
LM K AΓ
energy (eV)
2
0
- 2
π
µ
σ
graphite
-
π
σµ
LM K AΓ
MgB2
energy (eV)
0
2
4
2
FIG. 2. Schematic band structure of graphite (top panel)
and MgB2(bottom panel). Grey region in top panel indi-
cates the doping region achieved by chemical intercalation of
graphite (±1 eV).
These two compounds are indeed structurally and elec-
tronically very similar. A main difference is the relative
position of the σand πbands with respect to the chem-
ical potential µ. In undoped graphite the Fermi energy
cuts the πbands just at the K point, where the density
of states (DOS) vanishes. Doping graphite with donors
or acceptors, however, shifts the chemical potential µof
≃ ±1 eV providing metallic charges in the system and
a finite DOS [13]. This situation, on the other hand, is
naturally accounted in MgB2, where µlies well below the
π-band crossing at the K point and even crosses the two
σbands (see Fig. 2, where a pictorial sketch of the band
structure is drawn). Note that in the conventional ME
context the only electronic relevant parameter is just the
DOS at the Fermi level N(0). From this point of view
the difference between the superconducting properties of
MgB2with Tc= 39 K and intercalated doped graphite
with Tcup to 0.55 K at ambient pressure is hard to jus-
tify since both the materials show similar N(0). Such a
comparison suggests that the origin of the high-Tcphase
in MgB2should be sought among the features which dif-
ferentiate MgB2from doped graphite.
A similar impasse was encountered in the ME descrip-
tion of superconductivity in fullerenes, which also share
many similarities with graphite. Even there, LDA esti-
mates of the el-ph coupling λwere insufficient to account
for the high Tcand for the small isotope effect. Such
a discrepancy has been explained in terms of opening
of nonadiabatic channels which, under favourable con-
ditions fulfilled in fullerenes, can effectively enhance the
superconducting pairing [14]. A key role is played by the
small Fermi energy EFthat in fullerenes is of the same
order of the phonon frequency, violating the adiabatic
assumption (ωph ≪EF). In this situation Migdal’s theo-
rem [15], on which conventional ME theory relies, breaks
down. The proper inclusion of the nonadiabatic contri-
butions follows the framework of Ref. [16] and leads to a
new set of equations for superconductivity [17]:
Z(ωn) = 1 + Tc
ωnX
ωm
ΓZ(ωn, ωm, Qc)ηm,(1)
Z(ωn)∆(ωn) = TcX
ωm
Γ∆(ωn, ωm, Qc)∆(ωm)
ωm
ηm,(2)
where ηm= 2 arctan{EF/[Z(ωm)ωm]},Z(ωn) is the
renormalization function and ∆(ωn) is the supercon-
ducting gap function in Matsubara frequencies. The
breakdown of Migdal’s theorem strongly affects the “on-
diagonal” ΓZand the “off-diagonal” Γ∆el-ph kernels
which include now vertex and cross contributions [16]:
ΓZ(ωn, ωm, Qc) = λD(ωn−ωm)[1 + λP (ωn, ωm, Qc)],
Γ∆(ωn, ωm, Qc) = λD(ωn−ωm)[1 + 2λP (ωn, ωm, Qc)]
+λ2C(ωn, ωm, Qc)−µ,
where D(ωn−ωm) is the phonon propagator and µ
the dynamically unscreened Coulomb repulsion, to be
not confused with the chemical potential. The vertex
and cross functions, P(ωn, ωm, Qc) and C(ωn, ωm, Qc),
represent an average of the nonadiabatic diagrams over
the momentum space probed by the el-ph scattering,
parametrized by the quantity Qc.
In the nonadiabatic context outlined above, the role
of the σbands in MgB2acquires a new and interesting
perspective. Indeed the Fermi energy of these bands Eσ
F
is also quite small, Eσ
F∼0.4−0.6 eV [5], leading to
ωph/EF∼0.1−0.2. These values, together the sizable
λ∼1, point towards a similar size of the vertex cor-
rections λωph/EF∼0.1−0.2 and nonadiabatic channels
induced by the breakdown of Migdal’s theorem can be
therefore expected to be operative. In this situation it
is clear that the use of conventional ME framework can
lead to inconsistent results and a nonadiabatic approach
is unavoidable. The scenario we propose is the following:
•MgB2can be described as a multiband system with
two conventional ME bands π(with large Eπ
F>3 eV)
and two nonadiabatic bands σ(Eσ
F∼0.4−0.6 eV).
•πbands can be in good approximation can be consid-
ered as conventional. They could possibly contribute to
the dynamical screening of µ∗and to the static screening
(Thomas-Fermi like) of the long-range el-ph interaction.
They can also lead to the opening of a smaller super-
conducting gap in the πbands which does not probed
directly nonadiabatic effects.
2
0.0 0.2 0.4 0.6
Q=q/2kF
0.0
0.1
0.2
0.3
0.4
P(ωn,ωm,Q)
0.6 0.7 0.8 0.9 1.0
λ
0.5
1.0
1.5
2.0
2.5
Tc(vertex)/Tc(ME)
−1.1−0.6−0.10.4 0.9 1.4 1.9 2.4
k
−0.8
−0.6
−0.4
−0.2
0.0
ε(k)
(c)
(a)
(b)
µ
ω0
FIG. 3. (a) Momentum structure of the vertex function for
a parabolic 2D hole-like band. Different curves correspond to
different hole-fillings shown in panel (b). (c) Estimate of the
enhancement of Tcfor the nonadiabatic vertex theory with
respect to the ME one.
•High-Tcsuperconductivity is mainly driven by σ-
band states. The peculiar feature of such bands is the
smallness of the Fermi energy which induces new (nona-
diabatic) channels of el-ph interactions. Origin of the
high-Tcsuperconductivity is the effective enhancement
of the superconducting pairing as long as vertex correc-
tions result positive [P(ωn, ωm, Qc)>0].
As seen in the last item, an important element in this
scenario is the overall sign of the nonadiabatic effects,
which governs the enhancement or the suppression of Tc.
In previous studies we showed that the vertex function
Proughly obeys the simple relation [18,19]:
P > 0vFq/ω <
∼1
P < 0vFq/ω >
∼1,(3)
where ωis a generic exchanged energy involved in the
scattering of order of ωph , and vFis the Fermi veloc-
ity. In fullerene compounds, the strong electronic cor-
relation favours the small qmomentum el-ph coupling
[vFq/ω <
∼1] [20] probing therefore the positive part of
the vertex function P.
In MgB2the situation is deeply different. In fact the
nonadiabatic regime in MgB2is related to the closeness
of the Fermi level to the top of the 2D σ-bands, and
the non trivial dependence of the momentum-frequency
structure of Pon the filling has thus to be taken into
account [21]. To this regard, Eq. (3) is very helpful
to illustrate this point since, for parabolic hole bands,
vF∝p|µ|where µis the chemical potential with respect
to the top of the band. As µis made smaller, the positive
region of the vertex function will be enlarged and will
eventually cover the whole momentum space. Hence, in
MgB2the nonadiabatic vertex diagrams are intrinsically
positive in the whole momentum space regardless any
electronic correlation.
In Fig. 3 we show the numerical calculation of the
momentum structure (panel a) of the vertex function
P(ωn, ωm, Q) (Q=q/2kF) for different hole filling of
2D parabolic hole-like band(panel b). In panel a, the
exchanged energy ωn−ωmhas been set equal to ω0/2,
where ω0is an Einstein phonon representing the charac-
teristic phonon energy scale. The structure of the ver-
tex function is strongly dependent on the position of the
chemical potential. In particular for almost filled band
systems, as MgB2, the vertex structure becomes shape-
less and positive (solid lines). In such a situation the con-
tribution of the nonadiabatic vertex function is positive
in the whole momentum space, and nonadiabatic chan-
nels are expected to enhance Tcregardless the amount
of electronic correlation. This trend is shown in Fig. 3c
where the enhancement of Tcdue to nonadiabatic vertex
corrections is reported. The calculation of Tcfollows a
procedure similar to the one employed in Ref. [19], where
the vertex and cross functions are replaced by their re-
spective averages over the momentum transfer and by
setting ωn−ωm=ω0. Note how, as µmoves towards the
top of the band (panel b), Tcgets significantly enhanced
by the opening of nonadiabatic channels already for val-
ues of λconsistent with the LDA calculations. Similar
results were reported within the infinite dimensions ap-
proximation [22].
It should be noted that the almost 2D character is an
important ingredient for having a substantial value of Tc
because the density of states remains finite at the band
edge [5]. A 3D parabolic hole doped band would in fact
lead to a DOS proportional to p|µ|, which vanishes as µ
goes to zero. Additional effects can moreover arise from
an intrinsic momentum modulation of the el-ph interac-
tion. Low values of hole doping would in fact enlarge the
screening length leading to an el-ph interaction peaked
at small momentum transfer. A similar argument was
proposed for instance in relation to copper oxides [23]
and, in principle, it could explain the reflectance data in
MgB2[9,10]. Both the argumentations can of course hold
true and coexist in MgB2, explaining the high-Tcsuper-
conductivity in this material as effect of a nonadiabatic
el-ph pairing.
We would like to stress that, once σbands are ac-
cepted to play a key role in the superconducting pair-
ing of MgB2, nonadiabatic effects are unvoidably present
due to the smallness of their Fermi energy. The onset
of nonadiabatic channels can thus provide a natural ex-
planation for the inconsistency between the theoretical
values of λcalculated by LDA technique (λ≃0.7−0.9)
and the high value λ>
∼1.4 needed to reproduce experi-
mental data Tc= 39 K and α= 0.30.
Signatures of a nonadiabatic interaction can be found
however in other anomalous properties of MgB2. The
analysis of these features can provide further indipen-
dent evidences for the nonadiabatic pairing and suggest
precise experimental tests.
Impurities and chemical doping. A remarkable
3
reduction of Tcupon radiation-induced disorder has re-
cently been observed in MgB2[24], in contrast with An-
derson’s theorem. This kind of reduction in a s-wave
superconductor has been shown to be one of the charac-
teristic feature of a nonadiabatic pairing [25], as seen for
instance in fullerenes [26]. The experimentally observed
reduction of Tccan be therefore a further evidence of
nonadiabatic superconductivity. Similar conclusions can
be drawn by the analysis of the chemical substitutional
doping in MgB2. In fact, both electron [27] and hole [28]
doped materials show a lower Tcthan the pure stoichio-
metric MgB2. It is clear however that the contemporary
suppression of Tcupon electron or hole doping can not be
understood in terms of band filling. We suggest a much
more plausible scenario, namely that the stoichiometric
disorder induced by chemical substitution to be mainly
responsible for the reduction of Tc, with band filling as
a secondary effect. Again, since nonmagnetic ion substi-
tution does not break time reversal symmetry and An-
derson’s theorem in ME theory, a nonadiabatic pairing
appears as a natural explanation. To test this picture
the comparison with some completely substituted com-
pounds would be interesting.
Isotope effects. The detection of isotope effects on
various quantities receives a crucial importance in the
nonadiabatic framework since it directly probes the nona-
diabatic nature of el-ph interaction. In particular it has
been shown that nonadiabatic effects give rise to a finite
isotope effect on quantities which in conventional ME
theory are expected to not show it, for instance the ef-
fective electron mass m∗[29] and the spin susceptibility
χ[30]. The actual discovery of an anomalous isotope ef-
fects on these or other quantities represents therefore a
precise prediction of the nonadiabatic theory which could
be experimentally checked.
New high-Tcmaterials. Interesting suggestions can
come from the proposed nonadiabatic scenario in regard
to material engineering and superconductivity optimiza-
tion. According the analysis above discussed, a crucial
difference between low-Tcdoped graphite and high-Tc
MgB2is the upward shift of the σbands and their conse-
quent cutting of the Fermi level. The study of the relative
position of the σbands with respect of the πbands, and
of both of them with respect to the chemical potential
appears therefore extremely interesting. In particular
we would suggest that high-Tcsuperconductivity could
be achieved in MgB2-like materials when Fermi level is
lower but very close to the top of the σbands. On the
contrary we predict no high-Tcsuperconductivity in the
same family if compounds when i) Fermi level does not
cross the σbands, ii) or where the Fermi level is very
distant from the top of the σbands (EF>1 eV) and the
system looses its nonadiabatic nature. Theoretical calcu-
lations which can stimulate material engineering in this
sense are in progress. A potential candidate would be the
hole doped graphite as long as Fermi level could be low-
ered to cut the underneath σbands or the σbands arisen
by electrostatic effects. High level of chemical doping by
acceptor intercalation was for long time unsuccessful in
graphite as well as in C60 since such compounds resulted
unstable [13]. The recent discoveries of superconductiv-
ity at Tc= 35 K in graphite-sulphur compounds [31] and
at Tc= 117 K in FET hole-doped fullerenes [2] could
thus both arise from the unifying framework of the nona-
diabatic superconductivity. We thus encourage renewed
work along these lines.
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