Nonadiabatic breakdown and pairing in high-Tc compounds
The electron-phonon interaction plays a fundamental role on the superconducting and normal state properties of all the high-Tc materials, from cuprates to fullerenes. Another common element of these compounds is in addition the extremely small Fermi energy EF, which is comparable with the range ω...
Збережено в:
| Опубліковано в: : | Физика низких температур |
|---|---|
| Дата: | 2006 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2006
|
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/120187 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Nonadiabatic breakdown and pairing in high-Tc compounds / L. Pietronero, E. Cappelluti // Физика низких температур. — 2006. — Т. 32, № 4-5. — С. 455–478. — Бібліогр.: 141 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-120187 |
|---|---|
| record_format |
dspace |
| spelling |
Pietronero, L. Cappelluti, E. 2017-06-11T12:07:28Z 2017-06-11T12:07:28Z 2006 Nonadiabatic breakdown and pairing in high-Tc compounds / L. Pietronero, E. Cappelluti // Физика низких температур. — 2006. — Т. 32, № 4-5. — С. 455–478. — Бібліогр.: 141 назв. — англ. 0132-6414 PACS: 74.10.+v, 63.20.Kr https://nasplib.isofts.kiev.ua/handle/123456789/120187 The electron-phonon interaction plays a fundamental role on the superconducting and normal state properties of all the high-Tc materials, from cuprates to fullerenes. Another common element of these compounds is in addition the extremely small Fermi energy EF, which is comparable with the range ωph of the phonon frequencies. In the situation the adiabatic principle ωph/EF 1, on which the standard theory of the electron-phonon interaction and of the superconductivity relies, breaks down. In this contribution we discuss the physical consequences of the breakdown of the adiabatic assumption, with special interest on the superconducting properties. We review the microscopic derivation of the nonadiabatic theory of the electron-phonon coupling which explicitly takes into account higher order electron-phonon scattering not included in the conventional picture. Within this context we discuss also the role of the repulsive electron-electron correlation and the specific phenomenology of cuprates and fullerides. The authors acknowledge fruitful collaborations on this subject with C. Grimaldi, S. Strassler, P. Benedetti, M. Scattoni, P. Paci, M. Botti, L. Boeri, S. Ciuchi and G.B. Bachelet. We also acknowledge financial support from the MIUR projects COFIN03 and FIRB RBAU017S8R. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур General Aspects Nonadiabatic breakdown and pairing in high-Tc compounds Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Nonadiabatic breakdown and pairing in high-Tc compounds |
| spellingShingle |
Nonadiabatic breakdown and pairing in high-Tc compounds Pietronero, L. Cappelluti, E. General Aspects |
| title_short |
Nonadiabatic breakdown and pairing in high-Tc compounds |
| title_full |
Nonadiabatic breakdown and pairing in high-Tc compounds |
| title_fullStr |
Nonadiabatic breakdown and pairing in high-Tc compounds |
| title_full_unstemmed |
Nonadiabatic breakdown and pairing in high-Tc compounds |
| title_sort |
nonadiabatic breakdown and pairing in high-tc compounds |
| author |
Pietronero, L. Cappelluti, E. |
| author_facet |
Pietronero, L. Cappelluti, E. |
| topic |
General Aspects |
| topic_facet |
General Aspects |
| publishDate |
2006 |
| language |
English |
| container_title |
Физика низких температур |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| format |
Article |
| description |
The electron-phonon interaction plays a fundamental role on the superconducting and normal
state properties of all the high-Tc materials, from cuprates to fullerenes. Another common element
of these compounds is in addition the extremely small Fermi energy EF, which is comparable with
the range ωph of the phonon frequencies. In the situation the adiabatic principle ωph/EF 1, on
which the standard theory of the electron-phonon interaction and of the superconductivity relies,
breaks down. In this contribution we discuss the physical consequences of the breakdown of the
adiabatic assumption, with special interest on the superconducting properties. We review the microscopic
derivation of the nonadiabatic theory of the electron-phonon coupling which explicitly
takes into account higher order electron-phonon scattering not included in the conventional picture.
Within this context we discuss also the role of the repulsive electron-electron correlation and
the specific phenomenology of cuprates and fullerides.
|
| issn |
0132-6414 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120187 |
| citation_txt |
Nonadiabatic breakdown and pairing in high-Tc compounds / L. Pietronero, E. Cappelluti // Физика низких температур. — 2006. — Т. 32, № 4-5. — С. 455–478. — Бібліогр.: 141 назв. — англ. |
| work_keys_str_mv |
AT pietronerol nonadiabaticbreakdownandpairinginhightccompounds AT cappellutie nonadiabaticbreakdownandpairinginhightccompounds |
| first_indexed |
2025-11-25T20:40:24Z |
| last_indexed |
2025-11-25T20:40:24Z |
| _version_ |
1850526236623765504 |
| fulltext |
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5, p. 455–478
Nonadiabatic breakdown and pairing in high-Tc
compounds
L. Pietronero and E. Cappelluti
Dipartimento di Fisica, Universit� «La Sapienza», P. le A. Moro 2, 00185 Rome, Italy
INFM-CNR, SMC-Istituto dei Sistemi Complessi, CNR, v. dei Taurini 19, 00185 Rome, Italy
E-mail: Emmanuele.Cappelluti@roma1.infn.it
Received August 25, 2005
The electron-phonon interaction plays a fundamental role on the superconducting and normal
state properties of all the high-Tc materials, from cuprates to fullerenes. Another common element
of these compounds is in addition the extremely small Fermi energy EF, which is comparable with
the range �ph of the phonon frequencies. In the situation the adiabatic principle �ph/EF �� 1, on
which the standard theory of the electron-phonon interaction and of the superconductivity relies,
breaks down. In this contribution we discuss the physical consequences of the breakdown of the
adiabatic assumption, with special interest on the superconducting properties. We review the mic-
roscopic derivation of the nonadiabatic theory of the electron-phonon coupling which explicitly
takes into account higher order electron-phonon scattering not included in the conventional pic-
ture. Within this context we discuss also the role of the repulsive electron-electron correlation and
the specific phenomenology of cuprates and fullerides.
PACS: 74.10.+v, 63.20.Kr
Keywords: high-Tc superconductivity, electron-phonon interaction, fullerenes.
1. Introduction
For many years the concept of superconductivity
has been strictly associated with the electron-phonon
interaction. One of the fingerprints of the
phonon-based Migdal—Eliashberg (ME) theory in
conventional low-temperature superconductors has
been in fact the prediction and observation of many
peculiar features which are a direct evidence of a
phonon mediated superconductivity, for instance the
isotope effect �Tc
on the critical temperature, the ex-
traction of the electron-phonon (el-ph) coupling func-
tion � �2F( ) from tunneling experiments, phonon
anomalies occurring at temperature T Tc� , etc. [1].
The belief that superconductivity was intimately re-
lated to an electron-phonon pairing was so strong that
a semi-empirical upper limit for the critical tempera-
ture Tc
max � 20–25 K was thought to be valid before
the occurrence of lattice instabilities [2,3], in agree-
ment with the maximum Tc � 23 K achieved in
Nb3Sn.
This phonon-based scenario was shaken in 1986 by
the discovery of high-Tc superconductivity in copper
oxides [4] with Tc’s up to 140 K, well above the em-
pirical limit Tc
max � 20–25 K. In addition, the isotope
effect on the critical temperature at the optimal dop-
ing �opt in cuprates was found to be unconventionally
small �Tc
� 01. [5], suggesting a nonphonon mediated
mechanism. Following this perspective a large amount
of work has been devoted in the two last decades to
the study of purely electronic models to explain the
high-Tc superconductivity.
In the recent years however, the evidence of an im-
portant role of the electron-phonon interaction on
many properties of the normal and superconducting
state has been increasing. On one hand, the small
value of �Tc
turned out to be a peculiarity of the opti-
mal doping, whereas in the underdoped region �Tc
could be significantly larger, even higher than the
BCS limit �Tc
� 0 5. [6,7]. On the other hand a re-
markable isotope effect on the zero temperature Lon-
don penetration depth �L( )0 has been observed both
in the nearly optimal and in the underdoped regime
© L. Pietronero and E. Cappelluti, 2006
[8–10]. The finite isotope shift on �L( )0 has been re-
lated to an isotope effect on the effective electron mass
m* through the relation �L sn /m( ) *0 � , where n s
is the superfluid density. The observation of a finite
isotope effect on �L( )0 or on m* is highly puzzling
since these quantities are expected to show strictly
zero isotope effect in the conventional elec-
tron-phonon framework. The report of a finite isotope
effect on m* can be thus regarded not only as an indi-
cation of an important role of the electron-phonon
coupling, but also as an evidence of the unconven-
tional nature of the el-ph interaction [11–15]. Further
support to a significant electron-phonon coupling in
cuprates comes from angle-resolved photoemission
spectroscopy (ARPES). The kink of the electronic dis-
persion observed by these measurements was indeed
claimed to be of phononic origin, since it shows a neg-
ligible dependence on doping, on temperature and on
the angle direction along the Fermi surface [16].
Moreover an isotope shift, not only of the kink itself,
but also of the high energy electronic dispersion,
points out once more a dominant but still not well un-
derstood role of the electron-phonon interaction [17].
Motivated by this experimental scenario, there is
nowadays a revamping interest about the unconven-
tional role of the electron-phonon interaction in
cuprates and its relation with superconductivity. The
observation of high-Tc superconductivity with critical
temperatures up to Tc
max � 40 K in fullerenes [18–21]
and in the recently discovered MgB2 [22], where the
phononic origin of the superconducting pairing is
widely accepted, points out that a phonon-based
mechanisms can be actually a route for high-Tc super-
conductivity, and it suggests a common mechanism for
all these compounds.
An useful insight on this issue is provided by the so
called Uemura’s plot [23,24] (Fig. 1) which shows
that high-Tc and exotic superconductors (cuprates,
fullerenes, but also heavy fermions, perovskites and
Chevrel phases) are all characterized by a small den-
sity of charge carriers n, which is usually parametrized
in terms of the Fermi energy EF . This feature suggests
that a positive role can be played by the small carrier
density within an unique unconventional framework
relevant for all these superconductors. This hypothesis
could look quite puzzling because in conventional sys-
tems a low charge carrier density is usually considered
detrimental for superconductivity since it decreases
the number of metallic charges available for the Coo-
per pairing and it decreases the dynamical screening of
the electron-electron Coulomb repulsion.
In the following we shall identify in the small
Fermi energy EF induced by low values of n the miss-
ing link between small carrier density and high-Tc su-
perconductivity. More precisely we shall see that
when the electronic Fermi energy EF is small enough
to be comparable with the characteristic energy of a
generic boson mediator (�ph for the phononic case)
one of the basic assumption of the conventional
Migdal—Eliashberg theory, the adiabatic principle
�ph �� EF , breaks down [25]. This situation calls for
a generalization of the ME theory to explicitly include
nonadiabatic effects in the same spirit as self-energy
renormalization is taken into account in the ME the-
ory itself with respect to the BCS one [26].
The review is structured as the following: in Sec. 2
we introduce the adiabatic problem and we discuss the
implications arising from the failure of the adiabatic
assumption in terms of the quantum field theory. In
Sec. 3 we present the generalized theory of supercon-
ductivity in the nonadiabatic regime which takes into
account the onset of nonadiabatic channels of interac-
tion, while the relevance of the nonadiabatic scatter-
ing and the peculiar features of specific high-Tc mate-
rials, as fullerenes and cuprates, will be discussed in
Sec. 4 within the context of the nonadiabatic elec-
tron-phonon theory.
2. Breakdown of Migdal’s theorem
2.1. The nonadiabatic hypothesis
One of the most popular principles in solid state
physics is the adiabatic assumption which is on the ba-
sis of Born—Oppenheimer approximation [27]. Fun-
damental element of this approximation in metallic
systems is the observation that the electron energy
scale, parametrized by the Fermi energy EF , is some
orders of magnitude (typically 103–104) larger than
the lattice dynamics energy ruled by the phonon fre-
456 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
L. Pietronero and E. Cappelluti
❉❉❉
❉
❉❉
❄
❄
✦
Low Tc
MgAlB2 , MgCB2❉
A15
BKBO
A3 C60❄
BEDT✦
Chevrel
✜
✩
✩✩
✩
✩
La214
Y123
Bi2223
TMTSF✜
Heavy Fermions✩
2
–1
0
1
1 2 3 4 5
E [K]F
Zn
Al
Sn
Nb
10
10
10
10
T
[K
]
c
10 10 10 10 10
Fig. 1. Re-elaboration of Tc vs. EF plot after Refs. 23, 24
including magnesium diboride alloys.
quencies �ph. This situation leads to the decoupling of
the electron and lattice dynamics providing a suitable
approximation scheme where the complex many-body
problem becomes now affordable.
A powerful theoretical tool to deal with interacting
electron systems is the quantum field theory which
can be conveniently cast in terms of Feynman’s dia-
grams [28,29]. The diagrammatic Feynman’s approach
results particularly useful when nonperturbative
schemes are needed, as in the case of superconductiv-
ity, and in identifying classes of diagrams associated
with a particular physical property. Feynman’s theory
provides indeed an elegant way to generalize the BCS
theory in the strong coupling Migdal—Eliashberg re-
gime.
From a generic point of view all the physical prop-
erties of an interacting electron system could be com-
puted from the knowledge of the electron Green’s
function G or, in an equivalent way, of the electron
self-energy �, which is related to G through the
Dyson’s equation [28]:
G( , ) [ ( ) ( , )] .k k k� � ��
�� 1 (1)
The self-energy itself can be in its turn expressed as a
functional of the electron Green’s function G, of the
effective electron-electron potential V, and of the so
called vertex function �: � � �� [ , , ]G V . In the elec-
tron-phonon case here considered, or for any elec-
tron-electron interaction mediated by a generic boson,
the potential V can be directly related to the phonon
(boson) propagator D, so that
� � �� [ , , ].G D (2)
In the normal state Eq. (2) has the particular simple
form:
� �( ) ( ) ( ) ( ) ( , ),k dk g k k D k k G k k k k�
�
�
� � � �
� (3)
where the indexes k and k� comprehend both frequen-
cies and momenta and g is the electron-phonon matrix
element. A diagrammatic expression of Eq. (3) is
shown in Fig. 2, where the solid line represents the
electron propagator G, the wavy line the phonon
Green’s function D, and the filled circles the elec-
tron-phonon vertex function �. The complex many-
body nature of the problem is thus hidden in the un-
known quantity � which in principle does not have an
analytical expression. It is often useful to split the to-
tal vertex function into a zero-order constant term
plus a vertex correction function denoted as P:
�( , ) ( )[ ( , )] ,k k k g k k P k k k� �
�
�
� �
1 (4)
where P contains all the higher order interaction pro-
cesses and it is constituted by an infinite set of dia-
grams.
The evaluation of the electronic self-energy, as ex-
pressed in Eq. (3), still constitutes a formidable task
which cannot be analytically performed. A huge sim-
plification to this aim was provided in the late 50ies
by the so-called Migdal’s theorem [25] which showed
that, in the (adiabatic) limit �ph/EF � 0, the vertex
correction function P scales as
P
EF
� �
�ph
. (5)
In common metals, as above discussed, we have
�ph/EF � �10 3–10 4� . The total vertex function can
be thus safely replaced by its lowest order (constant)
term, � � 1 signalizing that the electron-phonon ver-
tex function is not renormalized by strong-coupling
effects in the adiabatic regime. In this framework the
normal state self-energy simply reads thus:
�( ) ( ) ( ) ( ).k dk g k k D k k G k�
�
�
� �� 2 (6)
In similar way a self-consistent equation for the
superconducting order parameter � can be derived un-
der the assumption of Migdal’s theorem validity [30]:
� �( ) ( ) ( ) ( ) ( ) ( ).k dk g k k D k k G k G k k�
�
�
� �
� �� 2
(7)
Equations (6) and (7), whose diagrammatic re-
presentation is shown in Fig. 3, define a closed set
Nonadiabatic breakdown and pairing in high-Tc compounds
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 457
�����
Fig. 2. Diagrammatic expression of the electron-phonon
self-energy. The solid line represents the electron Green’s
function, the wary line the phonon propagator and the
filled circles the electron-phonon vertex function.
�
a
�
� b� �
Fig. 3. Self-energy (a) and superconducting pairing (b)
of an electron-phonon system in conventional Migdal—
Eliashberg framework.
of equations provided the phonon spectral function
is known. They are thus the basis of the Migdal—
Eliashberg theory of the electron-phonon interaction
which was successful to describe many properties of
common metal and low-temperature superconductors.
For instance the normal state electron-phonon self-
energy at low energy simply reads:
� �
�� , (8)
and the coherent part of the Green’s function can be
written as
G
Z /Z
( , )
( )
,k
k
�
�
�
1 1 (9)
where Z is the electron-phonon renormalization factor
Z �
1 �. Migdal—Eliashberg theory leads thus to an
effective renormalization by the factor 1
� of many
physical quantities of the system, as the Fermi veloci-
ty v v /F F
* ( )�
1 � , the electron mass m m* ( )�
1 �
or the specific heat �� � ��
( ) ( )1 2 0 32N / [27].
Similar renormalization arguments permit to un-
derstand in a qualitative way the generalization from
the BCS equation for Tc,
Tc �
�
��
�
��
�
�ph exp ,
1
(10)
to the McMillan’s formula
Tc �
�
�
�
�
�
��
�
� �� �ph exp
. ( )
( . )
,
104 1
1 0 62
(11)
valid in strong coupling regime. Taking into account
the renormalization of the quasi-particle density of
states at the Fermi level N N*( ) ( )( )0 0 1�
� as well
as of the electron-phonon matrix elements
g g/* ( )�
1 � leads indeed to an effective elec-
tron-phonon coupling �� � ��
/( )1 which, once
plugged in Eq. (10), gives:
Tc �
�
��
�
��
�
�
�ph exp ,
1
(12)
which is nothing else that the strong coupling
McMillan’s formula in the absence of Coulomb repul-
sion �*.
It is interesting to note that at the adiabatic
ME level all the normal state quantities (renormalized
Fermi velocity vF
* , electronic mass m*, specific
heat ��, quasi-particle spectral weight Z, as well
as the reduced critical temperature t T /c c� �ph,
depend on the electron-phonon interaction only
through the parameter � which is independent
of the atomic mass Mat . The isotope coefficient
� A d A/d M�
log atlog on the normal state pro-
perties is thus expected to be zero while �Tc
� 0 5. .
We would like to make clear once more that all the
above results, commonly discussed within the frame-
work of the standard theory of electron-phonon inter-
action, strongly rely on the adiabatic assumption.
When this latter breaks down, a different theory with
qualitative different phenomenology is expected.
In order to give just the flavor of the possible deep
changes on the physical properties, let us schematize
the total electron-phonon vertex function � in non-
adiabatic regime as [31]
� �
�
�
�
�
�
�g
EF
1 �
�ph
, (13)
where the (nonadiabatic) nonzero-order vertex pro-
cesses were taken into account according with
Eq. (5). Equation (13) points out that a sizable
renormalization of the electron-phonon vertex func-
tion is expected in nonadiabatic systems. The effec-
tive phonon mediated electron-electron pairing �*
is thus modified in a schematic way in the form
� � �� �* ( ) ( )�
1 12
ph/E /F , and the correspond-
ing expression for Tc reads [31]:
T
/E
c
F
�
�
�
�
�
�
�
�
�
�
�
� ��
ph
ph
exp
( )
.
1
1 2 (14)
In the extreme nonadiabatic case �ph/EF � 1 and
assuming «normal» values � � 0 4. , �ph � 700 K,
Eq. (14) would thus easily give Tc � 135 K whereas
Eq. (12) would predict Tc � 24 K. We would like to
make clear once more that this simple minded gener-
alization should be considered only as indicative since
Eq. (5) estimates only the largest magnitude of first
order vertex diagrams, not their effective size or sign.
In order to determine in a more quantitative way how
the onset of nonadiabatic channels on interaction af-
fects the electron-phonon phenomenology, a more de-
tailed analysis is needed.
2.2. The vertex function beyond Migdal’s theorem
From the above discussion it is clear that a rigorous
way of evaluating the nonadiabatic effects is in princi-
ple not available since it would imply the full exact
solution of the electron-phonon many-body problem.
In this situation it is thus of the highest importance to
identify physical processes responsible for the non-
adiabatic phenomenology and to individuate the more
appropriate approaches to deal with them.
Along this view the polaronic picture [32–36] and
the nonadiabatic theory of superconductivity [37–39],
which we propose for the high-Tc compounds, describe
two distinct different contexts although they arise
from the same electron-phonon interaction [15]. This
difference is reflected in the different theoretical tool
458 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
L. Pietronero and E. Cappelluti
employed in the two cases. A perturbation approach is
for instance clearly inadequate to study the polaronic
state where ground-state properties are radically dif-
ferent from Fermi liquid ones [36,40]. Nontrans-
lationally invariant states are thus preferably used to
account for polaron localization [41,42]. On the other
hand a perturbative way appears a natural approach
when nonadiabatic effects are expected to modify the
Fermi liquid properties in a new nonadiabatic regime
without destroying its metallic character [37–39]. In
this perspective we believe that a perturbation ap-
proach sustained by a diagrammatic representation
can provide an useful insight on the relevant physical
processes in nonadiabatic systems and a qualitative in-
vestigation of their properties. On this basis a quanti-
tative description is therefore beyond the main pur-
poses of our work.
In generalizing the strong coupling theory of
Migdal—Eliashberg in nonadiabatic regime, a natural
starting point is suggested by Migdal’s theorem itself.
A controlled perturbative quantity is indeed identified
in Eq. (5) by the size of the vertex correction
��ph/EF , which can be therefore used as a small ex-
pansion parameter. First step of any perturbation
analysis of nonadiabatic effects is the explicit evalua-
tion of the first order vertex function P, diagrammati-
cally represented in Fig. 4. Its analytical expression
reads [25,26,37]:
P T g D
G
n m
l
n l
m
( , ; , ) | ( )| ( )
( ,
k k k p
k p k
p
� �
�
� �
��� � � �
�
2
� � �l n lG) ( , ),p (15)
where (k,�n), (k�,�m) and (p,�l) are, respectively,
the momenta and energies of the incoming, outcoming
and internal electrons in Matsubara notations, G and
D are the electron and phonon propagators and
g g( , ) ( )k p k p�
is the electron-phonon matrix ele-
ment.
The evaluation of the first order vertex diagram
beyond Migdal’s theorem [Eq. (15)] and its inclusion
in systems of electrons scattering with high-energy
bosons was previously addressed in the early 80ies
by Grabowski and Sham in the context of plas-
mon based superconductivity [43]. They estimated
the vertex function P in the static limit
lim lim ( , )q q� �0 0� �P where q k k� �
and
� � ��
m n are the exchanged momentum and fre-
quency, respectively. In this limit they found that the
nonadiabatic electron-phonon interferences described
by Eq. (15) disfavor the effective electron-boson at-
traction, namely lim lim ( , )q q� � �0 0 0� �P . Similar
results are found in later studies based on a local ap-
proximation, where the momentum dependence of the
vertex function is neglected in favor of averages over
the energy variables [44–50]. On this basis the inclu-
sion of vertex diagrams was assumed to be in any case
negative with respect to the superconducting pairing.
A full momentum dependent analysis shows however
that these results are just due to a partial analysis and
an effective enhancement of the electron-phonon in-
teraction can actually be induced when the complex
momentum structure of the vertex function is properly
taken into account [37–39].
In order to evidence this point it is useful to derive
an analytical expression of the vertex function which,
although approximated, can be used as a guideline to
discuss the general momentum-frequency structure.
To this aim we consider for the moment the most
simplified electron-phonon model containing however
all the essential ingredients to the problem. In par-
ticular we consider the first order vertex function
with bare electron and phonon propagators. Electrons
are assumed to interact with a dispersionless Ein-
stein phonon with energy �0 through a momen-
tum independent electron-phonon matrix element
g g( )k p
� . A half-filled constant density of
states will be in addition considered N N( )0 0�
[ ]
� �W W where the half-bandwidthW represents
the Fermi energy W EF� . Under these assumptions
Eq. (15) reads thus [26,37]:
P
g
T
i i
n m
n ll
m l
( , ; , )
( )
[
k k
p
� �
�
�
��� �
�
�
� � �
� �
2
1
2
0
0
2
0
2 2
�
i in l� � ( )][ ( )]
.
k p k p
(16)
To obtain an analytical expression the electronic expres-
sion can be further linearized around the Fermi level
Nonadiabatic breakdown and pairing in high-Tc compounds
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 459
k'
k – k'
p
p + k – k'
k
–
p
k
Fig. 4. Feynman’s representation of the first order dia-
gram appearing in the nonadiabatic regime.
��( ) ( ) ( ) ( ) | | cosp k k p v k k p q
�
� �
�
� p Fv
where � is the angle between p and q k k� �
, assuming
all the electronic momenta p, k, k� to lie on the Fermi
surface. The first order vertex diagram results thus
to be simply function of the exchanged momentum,
P Pn m n m( , ; , ) ( ; , )k k q� �� � � � :
P
N g
T dn m
n ll E
E
F
F
( ; , )
( )
( )
q � �
�
�
� � �
�
�
�
� �
�
�
2 0 2
0
0
2
0
2 2
1
1
2 2�
d
i i i E Q im l n F l
cos
( cos )( )
,
�
� � � � �
(17)
where E v kF F F� and Q kF� | |/q 2 . It can be shown
that the main frequency dependence of the vertex
function is only through the exchanged frequency
� �n m
. We can therefore set an external frequency
to zero �n � 0 so that the exchanged frequency �
reads just � � � ��
�m n m . We can now finally de-
rive the following analytic expression for P in the
limit of a small Q expansion valid for small ex-
changed phonon momenta [26,37]:
P Q
E Q
E
F
F
( , )� �
�
�
�
�
�
� �
�
�
�
��
�
� �
�
�
��
0
0 0
2
arctan arctan
�
�
�
�
�
�
�
�
�
�
�
�
�
�
��
� �
arctan
arctan
2
2
2
E Q
E Q
E Q
F
F
F
�
�
�
� � �
� �
�
�
�
�
�
�
�
�
�
� � �
� �
�
�
( )[( ) ]
[( ) ]
0 0
2 2
0
2 2 2
2E E
E
F F
F
�
��
.
(18)
Note that Eq. (18) provides an analytical derivation
of Migdal’s theorem by evaluating the limit
lim�0 0/EF
P� for generic values of Q and �. We ob-
tain:
lim ( , )
| |
�
� �
� � �
�0 0
0
02 2
2
/E FF
P Q
E Q
Q
�
�
!
""
#
$
%%
�
�
arctan�
�
�
� ,
(19)
which explicitly shows that the first order vertex dia-
gram scales as �0/EF in the adiabatic limit.
Equation (18) will be now used as basis to discuss
the complex momentum-frequency dependence of the
vertex function. In particular we are interested in de-
termining a possible increase or decrease of the effec-
tive electron-phonon pairing as due to the onset of
nonadiabatic effects. This concept can be quantita-
tively related to the sign of the vertex correction P,
where a positive sign (P & 0) leads to an enhancement
of the effective electron-phonon coupling and a nega-
tive sign (P � 0) to a reduction [51].
Positive and negative regions of the first order
vertex function P are plotted in Fig. 5 in the momen-
tum frequency Q—� space for an adiabatic ratio
�0 0 5/EF � . . We can see that the total sign of P in
not a priori defined but it depends strongly on the
specific values of Q and �. The complex momen-
tum-frequency structure of the vertex function P can
be characterized by its static and dynamic limits, re-
spectively, P s and Pd [37,51]. It can be shown in full
generality that:
P P Q
P P Q
s
Q
d
Q
� �
� &
� �
� �
lim lim ( , ) ,
lim lim ( , ) ,
0 0
0 0
0
0
�
�
�
�
(20)
(21)
signalizing a nonanalytic point of the vertex function
P in (Q � 0, � � 0).
We can now fully understand how the negative
sign found in Ref. 43 does not represents the total
structure of the vertex diagram but only a limit
(static) case. The evaluation of the effects of the
nonadiabatic vertex diagrams on the electron-phonon
coupling appears thus much more complex and it will
depend in general on the specific momentum-fre-
quency region actually probed by the electron-phonon
scattering in a particular material.
On one hand, in systems with negligible momentum-
frequency dependence of the electron-phonon inter-
action we can expect for instance that the wholeQ—�
space will be effectively probed. Positive and negative
parts of the vertex diagram in this case almost cancel
out with a slight dominance of the negative sign. A re-
sulting small reduction of the electron-phonon pairing
is thus expected, as confirmed by the analysis in
Refs. 44–50. On the other hand materials characte-
460 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
L. Pietronero and E. Cappelluti
0.2 0.4 0.6 0.8 1.0
� �
0
0
0.2
0.4
0.6
0.8
Q
V
P < 0
V
P > 0
Fig. 5. Sign of the vertex function in the Q—� space for a
nonadiabatic system (�0 05/EF � . ).
rized by a net predominance of forward (small q) scat-
tering would mainly probe the smallQ positive region
of P leading to an effective enhancement of the elec-
tron-phonon pairing [37–39].
The crucial role of the electron-phonon momentum
structure appears now evident from the above discus-
sion. In this context, approaches as the dynamic mean-
field theory (DMFT) or numerical calculations on
small clusters are expected to underestimate the im-
portance of the q dependence and are not suitable for a
correct evaluation of the nonadiabatic effects.
A useful tool to study in a controlled way the role
of possible q (Q) momentum selection in the non-
adiabatic framework is the introduction of a fictitious
cut-off qc (Q q / kc c F� 2 ) which selects small | |q � qc
momenta [37–39]. In next sections we shall related on
a more compelling ground the cut-off qc to the micro-
scopic properties of the system, e.g., the electronic
correlation [52–57]. The electron-phonon matrix ele-
ments can be thus modelled as [38]:
| ( )| ( ) | | .g f q g qc cq q2 2�
�' ( (22)
The prefactor f qc( ) can be properly chosen in order
to have a total electron-phonon coupling � indepen-
dent of qc, so that the effect of the small momenta se-
lection on the vertex function can be separated from
the total phase space reduction which would affect �.
The prefactor f qc( ) depends thus in principle on the
spatial dimension and on peculiar characteristics of
the model. For a isotropic system in three dimensions
we would have for instance [38]:
�| ( )| ,g Q
g
Q
Q Q
c
c
2
2
2
�
� (23)
where we have expressed the exchanged momentum
| |q in the dimensionless variable Q kF� | |/q 2 . Equa-
tion (23) ensures that the Fermi surface average of
the phonon mediated electron-electron interaction is
constant ) * �| ( )|g Q gFS
2 2. In this way the electron-
phonon coupling constant � and the cut-off parameter
Qc can be considered as two independent free vari-
ables.
On the basis provided by Fig. 5 and by Eq. (18) we
can now qualitatively investigate the role of a small
momenta selection in the context of a nonadiabatic
electron-phonon interaction. This issue will be
parametrized as function of the cut-off Qc, where
Qc � 1 represents a shapeless electron-phonon interac-
tion in the momentum space andQc �� 1 a marked for-
ward scattering predominance which could be repre-
sentative of strongly correlated systems [52–57]. In
order to evaluate the effect of the inclusion of the first
order vertex function P in a generic phonon mediated
property (electronic self-energy, superconducting
pairing) we consider a weighted average of P Q( , )� in
the whole momentum-frequency space, P Qc
av ( ) [37].
Weighting factors will be the momentum dependence
electron-phonon matrix elements as given by
Eq. (23), which would take into account a possible
small momentum selection, and a simple Lorentzian in
frequency � � �0
2
0
2 2/( )
to simulate the presence of a
phonon propagator with characteristic energy �0
which would be always connected to the elec-
tron-phonon vertex function.
The dependence of the weighted average vertex
function on the adiabatic ratio �0/EF is shown in
Fig. 6 for different values of the cut-off Qc. Note the
strong enhancement for small values ofQc and the fact
that a significant magnitude of P Qc
av ( ) is already ap-
preciable for relatively small values of �0/EF .
According to this preliminary qualitative analysis
we can then identify a physical regime characterized
by a finite adiabatic ratio �0 0/EF + and small mo-
mentum phonon scattering (Qc � 1) where the non-
adiabatic effects induced by the breakdown of Mig-
dal’s theorem could lead to a remarkable enhancement
of the electron-phonon pairing. Guided by these re-
sults a rigorous generalization of the theory of super-
conductivity in nonadiabatic regime [38,39] will be
presented in the next section. We are going to see that
the enhancement of the superconducting pairing sug-
gested by the above discussion is actually confirmed
by a numerical accurate analysis. The normal and
superconducting state phenomenology of the non-
adiabatic systems will be also investigated.
Nonadiabatic breakdown and pairing in high-Tc compounds
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 461
0 0.2 0.4 0.6 0.8 1.0
�
0
/E
F
0
0.1
0.2
0.3
0.4
P
av
(Q
c
)
Q c = 0.1
Qc = 0.9
Fig. 6. Momentum-frequency average of the vertex dia-
gram P Qc
av( ) as function of the adiabatic parameter
�0/EF for different values of the momentum cut-off Qc
(from the top to the bottom): Qc � 0.1, 0.3, 0.5, 0.7, 0.9,
and for � � 1.
3. Nonadiabatic theory of superconductivity and
normal state
In Sec. 2 we have briefly mentioned the possibility
to build a perturbative theory in regime of strong cou-
pling � , 1 on the basis of a small �0/EF expansion.
The most straightforward way to achieve this aim is
the use of the functional formalism based on the
Baym–Kadanoff technique [58,59] to derive a con-
serving scheme which links the superconducting and
normal state properties to a self-consistent evaluation
of the normal state self-energy in nonadiabatic regime.
In terms of Feynman’s diagrams a convenient starting
point is the skeleton formalism where each graphical
element represents fully renormalized quantities
[28,29].
3.1. One-particle self-energy
In this framework the diagrammatic generalization
of the electron-phonon self-energy at the first order in
a �0/EF expansion is depicted in Fig. 7. Retaining
only the first term on the right side is equivalent
to the so called noncrossing approximation (NCA)
where nonadiabaticity would be taken into account
only through finite bandwidth effects. More impor-
tance is attached to the second term on the right side
which explicitly contains the first order vertex dia-
gram arising from the breakdown of Migdal’s theo-
rem. Its analytic expression can be written in the com-
pact form as [11,38,39]:
�( , ) ( )| ( )|
[ ( , ; , )
,
k k k
k k
k
� � �n
m
n m
n m
T D g
P W W
�
� �
�
�
�
� 2
1 ] ( , ) ,G Wmk� (24)
where we have introduced the renormalized Matsu-
bara frequencies iW in n n�
� ��( ), D is the generic
phonon propagator coupled to the electrons through
the electron-phonon Eliashberg function � �2F( )
[27,60,61]:
D
F
dn m
n m
( )
( )
( )
,� �
�
�� �
� � �
�
�
�
2 2
2 2
(25)
and P is the first order vertex function:
P W W T g D
G W
n m
l
n l
l
( , ; , ) | ( )| ( )
( ,
,
k k k p
p k k
p
� �
�
�
�
�
�
2 � �
n m lG W� ) ( , ).p (26)
In an isotropic system the angular dependence
of observable quantities is negligible and it can be
dropped. We can therefore replace the self-energy in
Eq. (24) and the whole electron-phonon matrix ele-
ments in Eq. (26), including the vertex corrections,
with their averages over the Fermi surface [11,38,39]:
� � �( , ) ( , ) ( ),k k� � �n n FS n� ) * � (27)
| ( )| [ ( , ; , )]
[ ( , ; , )]
g P W W
P W W
n m
n m FS
k k k k
k k
�
� �
� )
� *
2 1
1 �
�
g P Q W Wc n m
2 1[ ( ; , )] . (28)
Equations (24), (27) and (28) define a self-consis-
tent expression for the self-energy which can be writ-
ten in the compact form:
W T P Q W W
D
E
W
n n
m
c n m
n m
F
M
�
�
�
!
""
�� �
� �
2 1[ ( ; , )]
( ) arctan
#
$
%% , (29)
and which can be numerically solved once the mo-
mentum averaged vertex function P Q W Wc n m( ; , ) is
known. An explicit expression of P Q W Wc n m( ; , ) can
be derived within the same approximations employed
in Sec. 2. However, because of the self-consistent
renormalization of the internal Green’s functions in
Eq. (26), we cannot obtain now an analytical form
for P Q W Wc n m( ; , ) since it would involve a sum over
the Matsubara frequencies. A longsome derivation of
P Q W Wc n m( ; , ), which is essentially based on a small
Qc expansion, can be found in Refs. 62, 63 where we
refer for more details. We report here therefore only
the final result:
462 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
L. Pietronero and E. Cappelluti
���
Fig. 7. Nonadiabatic electron-phonon self-energy including
the first order vertex diagram arising from the breakdown
of Migdal’s theorem.
P Q W W T D B n m l
A n m l B n m l
c n m
l
n l( ; , ) ( ) ( , , )
( , , ) ( , , )
�
�� � �
[ ]
( )
W W
E Q
l l n m
F c
�
!
"
"
"
� �
2
2 22
�
!
"
"
#
$
%
%
� � � �
1
4
1
1
2
1
42 2 2E Q
W W
E Q
W W
F c
l l n m
F c
l l n
ln
m
!
"
"
#
$
%
%
�
�
�
�
�
�
�
�
�
�
-
.
/
0
/
1
2
/
3
/
#
$
%
%
%%
2
1
2
, (30)
where
A n m l W W
E
W
E
Wl l n m
F
l
F
l
( , , ) ( )�
!
""
#
$
%%
� �
�
arctan arctan
n m�
!
""
#
$
%%
�
�
�
�
�
� , (31)
B n m l W W
E W
E W
E
E W
l l n m
F l n m
F l n m
F
F
( , , ) ( )
[ ]
�
� �
� �
� �
2 2 2 2
l n m� �
2
. (32)
Equations (29)–(32) define now a closed set of
equations which determine in an unambiguous way all
the one-particle properties of a nonadiabatic system.
The nonadiabatic equations are valid for any generic
electron-phonon spectrum described by the Eliashberg
function � �2F( ). They will be thus the basis for inves-
tigating the characteristic features in the normal state
of nonadiabatic systems. For practical purposes we
shall consider in the following, unless better specified,
a single Einstein phonon spectrum, which reproduces
however all the relevant nonadiabatic characteristics.
As we are going to see, interesting new physics is in-
duced by the onset of new channels of interaction in
the nonadiabatic regime.
A natural quantity to be investigated in this frame-
work is the effective electronic mass m* which is
renormalized by the electron-phonon interaction.
Hallmark result of the conventional Migdal—Eliash-
berg theory is the simple relation between the ef-
fective mass m* and the unrenormalized one m:
m m* ( )�
1 � [27]. Straightforward consequence of
this relation is the absence of any isotope effect on m*,
since the electron-phonon coupling constant � can be
shown not to depend on the atomic masses. This pre-
diction is indeed verified in all common metals.
The effective mass m* in our context can be easily
obtained from Eqs. (29)–(32) as a simple limit of the
renormalization function Z i i /in n n( ) ( )� � ��
1 �
[11]:
m
m
Z i
i
ii
n
i
n
nn n
*
lim ( ) lim
( )
.� �
�
�
�
�
�
�
� �� �
�
�
�0 0
1
�
(33)
In Fig. 8 we plot the quantity Z i n( )� , calculated
in nonadiabatic regime (�0 0 2/EF � . ) by including
the first order vertex diagram as in Fig. 7, as function
of �n for � � 10. [11]. A marked dip around �n � 0 is
observed in the nonadiabatic theory (solid lines)
which is absent when the vertex diagram is omitted
(dashed line, noncrossing approximation). This fea-
ture can be understood by considering that at �n � 0
the electron mass renormalization factor Z is mostly
modified by the static limit of the vertex function,
lim ( ; , )W W c n mm n
P Q W W� , which was previously
shown to be negative.
Another interesting feature appearing in the non-
adiabatic regime is an effective dependence of the elec-
tronic mass m* on the adiabatic ratio and conse-
quently on the atomic mass Mat . This concept leads
thus to a generalization of the standard Mig-
dal—Eliashberg expression for m* in nonadiabatic
systems:
m mf
Em
F
* , ,*�
!
""
#
$
%%�
�0 (34)
Nonadiabatic breakdown and pairing in high-Tc compounds
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 463
0 0.5 1.0 1.5 2.0 2.5
�n �0
1.1
1.3
1.5
1.7
1.9
Z
(i
�
n
)
Q c = 0.1
Q c = 0.5
Fig. 8. Renormalization function Z i n( )� for �0 02/EF � .
and � � 10. . Solid lines: nonadiabatic theory with
Qc � 01 02 05. , . , , .� . Dashed line: noncrossing approxima-
tion with no vertex diagram.
where lim ( , )*� � � �
0 0 0 1/E m FF
f /E� �
. The ex-
plicit dependence of the effective electronic mass
on the adiabatic ratio is reflected in a finite isotope
effect on m*. In Fig. 9 we report the isotope coef-
ficient � m* as function of the adiabatic parameter
�0/EF [11]. Solid lines correspond to the nonadia-
batic theory for different values of Qc, dashed line to
the noncrossing approximation where nonadiabaticity
was retained only through finite bandwidth effects.
A finite negative isotope effect is recovered in the
nonadiabatic regime, mainly due to the onset of
nonadiabatic channels of interaction (compared with
the dashed line). These results were later confirmed
by DMFT calculations [15]. Note moreover that � m*
does not show in the whole region of �0/EF a crucial
dependence on Qc. This is again due to the particular
limit of the vertex function probed by the electronic
mass m* which is not significantly affected by the
small | |q selection. The prediction of a finite and nega-
tive isotope effect on m* is of the highest importance
with respect to the experimental observation of an
isotope effect on the penetration depth �L( )0 in
cuprates which is indeed thought to stem from a cor-
responding isotope effect on the electronic mass m*.
3.2. Superconducting instability
The study of the nonadiabatic effects on the super-
conducting instability and on the resulting critical
temperature Tc appears of fundamental importance in
the light of the specific properties of the high-Tc ma-
terials, that we have seen to present small Fermi ener-
gies, and of the previous qualitative discussion
in Sec. 2 which suggested that an enhancement of
the superconducting pairing is achievable in the
nonadiabatic regime under favorable conditions like
the predominance of forward scattering [37].
The formal derivation of the nonadiabatic theory of
superconductivity follows essentially the same proce-
dure developed for the normal state. The explicit
equations can be determined by the requirement of a
conserving theory consistent with the self-energy � de-
picted in Fig. 7. Technical steps are the introduction
of an external field h� coupled with the superconduct-
ing condensate � , ) *c c† † and the functional deriva-
tive of � with respect to h� in the spirit of the
Baym—Kadanoff formalism.
The practical derivation is accomplished in the
clearest way in terms of the diagrammatic representa-
tion. The graphical expression of the superconducting
instability equation is thus obtained by replacing
in all the possible combinations one of the electronic
lines of Fig. 7 with an anomalous propagator which
can be in its turn expressed as function of the su-
perconducting order parameter � and of two normal
state propagators with (k, ,�n � ) and ( , , )
4k �n
quantum numbers [38,39], respectively. We obtain
then a self-consistent equation for the superconduct-
ing order parameter � as diagrammatically shown in
Fig. 10.
We can see that the failure of Migdal’s theorem
gives rise to two vertex diagrams, which have been al-
ready widely discussed, and to one so-called «cross»
term [38,39]. In similar way with the normal state
self-energy, we can write down the explicit self-con-
sistent equation for �:
� � � �( , ) ( )| ( )|
( , ; ,
,
k k k
k k
k
n n m
m
n m
T D g
P W W
�
� �
�
�
�
� 2
1 2[ �) ( , ; , )
( , ) ( , ) ( , ) ,
]
� �
� �
�
�
C W W
G W G W
n m
m m m
k k
k k k� � (35)
where the cross function C is defined as:
464 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
L. Pietronero and E. Cappelluti
� 0 /EF
0 0.2 0.4 0.6 0.8 1.0
0
–0.1
–0.2
–0.3
Q c = 0.6
Q c = 0.2
�
m
Fig. 9. Isotope coefficient �m* on the effective electronic
mass m* calculated for � � 1. Solid lines: nonadiabatic the-
ory for Qc � 02 03 06. , . , , .� . Dashed line: noncrossing ap-
proximation with no vertex diagrams.
� �
��
� �
���
Fig. 10. Self-consistent equation for the superconducting
order parameter � in the nonadiabatic theory.
C W W T g g
D D
n m
l
n l
( , ; , ) | ( )| | ( )|
( ) (
,
k k k p p k
p
� �
� �
�
� 2 2
� � � �l m
l n m lG W G W
�
�
� � �
)
( , ) ( , ) .p k k p (36)
For an isotropic superconductivity (we shall dis-
cuss later d-wave symmetry), we can also average the
superconducting order parameter � on the Fermi sur-
face. We end up with the resulting equation for the
superconducting instability in Matsubara frequencies:
�
� �
� �
�
n c c n m
m
n m c n m
m
T P Q W W
D C Q W W
W
�
�
�
�2 1 2[ ( ; , )]
( ) ( ; , )
m
F
m
E
W
arctan
!
""
#
$
%% .
(37)
The explicit derivation of the cross function can be
carried on in similar way with the vertex P to ob-
tain [62]:
C Q W W T D D
B n m l
c n m c
l
n l l m( ; , ) ( ) ( )
( , , )
�
�
�
�� � � � �2
2 arctan
4 2E Q
W W
A n m l B n m
F c
l l n m
!
"
"
#
$
%
%
�
-
.
/
0/
�
� �
( , , ) ( , ,l W W
E Q W W
l l n m
F c l l n m
)[ ]
[ ]
,
1
2
/
3/
� �
� �
2
2 22
(38)
where the functions A n m l( , , ) and B n m l( , , ) are the
same ones defined in Eqs. (31), (32).
Equation (37), together with Eq. (29), evaluated
at T Tc� , defines the nonadiabatic theory of supercon-
ductivity which can be employed for numerical calcu-
lations. We are now in the position to investigate with
a full numerical solution the role of the opening of
nonadiabatic pairing channels on the superconducting
critical temperature.
In Fig. 11 we show the critical temperature Tc as
function of the adiabatic parameter �0/EF for differ-
ent values of the dimensionless momentum cut-off Qc
and for � � 0 7. . The case �0 0/EF � corresponds to the
Migdal’s limit. Interestingly, nonadiabatic effects can
affect in different ways the superconducting pairing,
resulting in an increase or reduction of Tc depending
on the microscopic details of the system (the ratio
�0 0/EF � , the degree of the electronic correlation
Qc, ...). We can note however that the intuitive idea
that strong electronic correlations (small Qc’s) can fa-
vor superconductivity by selecting positive regions
of the vertex function is sustained by the numerical
calculations [38,39]. In Fig. 11 a marked enhance-
ment of Tc with respect to the conventional
Migdal—Eliashberg theory is shown for relatively
small values of Qc.
We can also easily compute from Eqs. (29), (37)
the isotope coefficient on Tc. The results are shown in
Fig. 12. We can note some interesting features. First
of all a reduced isotope effect �Tc
� 0 5. can be found
as result of the nonadiabatic pairing (note that in the
present analysis no Coulomb repulsion was taken into
account so that should be strictly equal to 0 5. in the
Migdal—Eliashberg framework). At the same time an
isotope coefficient �Tc
& 0 5. can be also observed,
which is a quite unconventional feature. It is interest-
Nonadiabatic breakdown and pairing in high-Tc compounds
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 465
0.1 0.2 0.3 0.4 0.5
0� /EF
0
0.05
0.10
0.15
0.20
0.25
T c
/�
0
cQ = 0.1
cQ = 0.9
� = 0.7
Fig. 11. Superconducting critical temperature Tc in the
nonadiabatic theory as function of the ratio �0/EF for
� � 07. and different values of Qc (from the top to the bot-
tom line): Qc � 01 03 05 07 09. , . , . , . , . .
0 0.1 0.2 0.3 0.4 0.5
�0 /E F
0.2
0.3
0.4
0.5
0.6
0.7
�
T
c
cQ = 0.1
cQ = 0.9
� = 0.7
Fig. 12. Isotope coefficient �Tc
on Tc in the nonadiabatic
theory as function of the adiabatic ratio. Same values of �
and Qc as in previous figure. Smaller values of Qc corre-
spond to lines with steeper initial slope.
ing to note also that the stronger variations of �Tc
are
predicted for small Qc’s, i.e., when higher Tc’s are re-
covered.
The Nambu formalism permits to generalize the
nonadiabatic theory of superconductivity also below
the critical temperature Tc to evaluate zero tempera-
ture quantities as for instance the superconducting
gap 5. The formal derivation is not difficult but quite
tedious and we refer to Ref. 64 for technical details.
An important point is that the vertex and cross func-
tions, as well the corresponding nondiagonal quanti-
ties in the Nambu space, need to be evaluated in the
presence of the superconducting gap which partially
reduces the nonanaliticity of these function at the
(q � 0, � � 0) point. The moment-frequency structure
of the vertex function in the superconducting state is
report in Fig. 13 which shows that the net result of the
opening of the superconducting gap is to reduce the
positive region of the vertex function and to increase
the negative one. A direct consequence of this effect is
that the effective superconducting pairing in the
superconducting state at T � 0 is smaller than the one
at T Tc� , so that the ratio 25/Tc, for a given �, is
smaller in the nonadiabatic framework than in ME
theory [64].
The scenario here outlined shows the breakdown of
the conventional picture of the superconducting and
normal state properties in nonadiabatic systems. In
particular it is clear that the inclusion of higher order
vertex terms arising from the violation of Migdal’s
theorem does not lead to an effective renormalization
of the electron-phonon coupling � but defines a new
physical regime where conventional microscopic pa-
rameters (�, �0, ...) can give rise to unconventional
features [65]. This is evident for instance by looking
at the isotope effects on Tc and on m* which would be
�Tc
� 0 5. and � m* � 0, respectively, if just a renor-
malization of � would be involved. This is clearly
shown also by the comparison between Figs. 9 and 12
where it is clear that the intrinsic dependence on the
microscopic parameters (�, �0,Qc, ...) can be very dif-
ferent according which physical quantity is consi-
dered.
Additional evidence of the breakdown of the con-
ventional phenomenology in nonadiabatic supercon-
ductors is provided by the anomalous dependence of
the critical temperature on nonmagnetic impurities
[62]. A well-known theorem in the conventional the-
ory of superconductivity states indeed that scattering
from disorder or nonmagnetic impurities should have a
strictly zero effect on Tc in isotropic s-wave supercon-
ductors in adiabatic regime [66]. A violation of this
theorem however has been experimentally observed in
fullerides and in electron-doped s-wave cuprates. The
nonadiabatic theory of superconductivity provides a
natural framework to explain this anomalous behavior
as shown in Fig. 14 where we show the strong reduc-
tion of the critical temperature Tc as function of the
impurity scattering rate � [62]. In agreement with the
previous data on Tc, m*, �Tc
, � m* , this unconven-
tional reduction is more marked for small Qc’s where
nonadiabatic effects are shown to be stronger.
466 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
L. Pietronero and E. Cappelluti
0 1 2
�/�0
0
1
–1
2
P
(Q
,
�
)
0 1 2 3
�/�0
Q = 0
Q = 1.0
� = 0
Q = 0
Q = 1.0
� = 0.1
Fig. 13. Frequency structure of the vertex function for
different values of the exchanged momentum: (from top
line to the bottom) Q � 0 02 04 10, . , . , , .� . The adiabatic pa-
rameter is here set �0 02/EF � . . Left panel refers to the
normal state (� � 0), right panel to the superconducting
state (� � 01 0. � ). Filled circles mark the static and dy-
namic limits in the normal and superconducting state.
0 0.2 0.4 0.6 0.8 1.0
/� 0
0.05
0.08
0.11
0.14
0.17
0.20
T c
/
�
0
� = 0.7
Q = 0.1c
Q = 0.3c
Q = 0.5c
�0 F/E = 0.2
Fig. 14. Critical temperature as a function of the impurity
scattering rate � for different values of Qc in the
nonadiabatic theory. The dashed line corresponds to the
noncrossing approximation with no vertex contribution.
3.3. Phenomenology in normal state: the Pauli spin
susceptibility
In the previous section we have stressed the impor-
tance to predict electron-phonon anomalous effects in
the nonadiabatic scenario which are not expected
within the ME theory and which can be employed as
direct experimental tests. One of them was the the ef-
fective electron mass m* which at the ME level de-
pends only on � but not on the phonon frequencies so
that no isotope effect should appear. Along this line
the Pauli susceptibility 6P is another promising quan-
tity to evidence nonadiabatic electron-phonon effects
since it is expected to be completely unaffected by the
electron-phonon coupling within the conventional
Migdal—Eliashberg theory [27]. Following the
Baym—Kadanoff formalism [58,59] one can write a
general expression for 6P :
6 � �P B
n
n nT G W W�
�2 2 2
k
k k
,
( , ) ( , ),� (39)
where �B is the Bohr magneton and �� is the total
spin vertex function. It can be shown that the elec-
tron-phonon interaction in the Green’s functions G
and in the spin vertex �� cancels out in the adiabatic
limit since the electron-phonon self-energy � ep in the
presence of a magnetic field H has nonzero contribu-
tions only at a nonadiabatic level [67–69]:
lim ( ).
H
ep
FHO /E
�
�
0
0� � (40)
Hence, any evidence of electron-phonon effects on the
Pauli spin susceptibility would be a direct proof of a
nonadiabatic electron-phonon coupling [63,69].
The proper inclusion of the electron-phonon inter-
action in Eq. (39) requires the evaluation of the spin
vertex function �� in nonadiabatic regime [63]. This
can be done following the diagrammatic scheme previ-
ously discussed for the two particle superconducting
response. The pictorial expression of the spin vertex
�� including electron-electron exchange interaction
and the nonadiabatic electron-phonon coupling is
shown in Fig. 15. Explicit expressions for the vertex
and the cross diagrams appearing in Fig. 15 can be
derived in similar way as in the Cooper pairing case,
by using the vertex and cross functions defined in
Eqs. (30), (38). Numerical calculations can be there-
fore performed by the self-consistent solution of the
Fig. 15 and of the frequency renormalization equation.
In Fig. 16 we plot the total spin susceptibility
(electron-electron + electron-phonon scattering) as
function of the electron-phonon coupling and of the
adiabatic parameter �0/EF for zero temperature [63].
Dashed lines are the results obtained within the non-
crossing approximation without electron-phonon ver-
tex diagrams [69], while solid lines are the data for
the fully vertex corrected theory [63]. For this latter
case we show the results for different values of Qc.
The first main results of Fig. 16 is that the inclu-
sion of the electron-phonon coupling, in the non-
adiabatic regime, yields a sensible reduction of 6 with
respect to the pure electronic spin susceptibility 6 ee .
As expected this effect vanishes as � � 0 (right panel)
or �0 0/EF � ( left panel). Note that both the non-
crossing approximation and the vertex corrected the-
ory yield a similar reduction. This is quite different
Nonadiabatic breakdown and pairing in high-Tc compounds
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 467
��
����
�����
Fig. 15. Diagrammatic representation of the spin vertex
function �� in nonadiabatic regime. Wavy lines represents
the electron-phonon interaction, dashed ones the elec-
tron-electron Coulomb repulsion.
0 0.4 0.8
� 0 /E F
0.5
0.6
0.7
0.8
0.9
1.0
�
/�
ee
� = 0.7
0 0.4 0.8
�
� 0 /E F = 0.7
Fig. 16. Spin susceptibility as function of the adiabatic
parameter �0/EF and of the electron-phonon coupling �.
The total Pauli spin susceptibility is normalized with re-
spect to the purely electronic one ee with a Stoner factor
N I( ) .0 04� . Dashed lines represent the spin susceptibility
in the noncrossing approximation, solid lines are the non-
adiabatic theory with vertex diagram (from lower to upper
line: Qc � 01 03 05 07 09. , . , . , . , . ).
from the situation encountered in the superconducting
pairing channel, where the effect of the vertex dia-
grams is much stronger and highly dependent on Qc
[38,39].
A more clear signature of the nonadiabatic effects is
in addition provided by the isotope dependence of the
Pauli spin susceptibility. In Fig. 17 we report the nu-
merical calculations of the isotope coefficient
� 6 6 �� �
�d /d M / d /dlog log ( ) log logat 1 2 0 as
function of the adiabatic ratio �0/EF and of �
[63,69]. A finite and negative isotope effect is thus
predicted with a strong dependence onQc and � in the
intermediate nonadiabatic regime. The observation of
the isotope effect, which would be absent in metals
fulfilling the Migdal—Eliashberg framework, could
be therefore an additional and stringent evidence for a
nonadiabatic electron-phonon coupling.
3.4. Photoemission and real axis analysis
In Secs. 3.1–3.3 we have briefly discussed the
nonadiabatic effects on some one- and two-particle
normal state properties. This analysis was simplified
by the fact that all these quantities (the effective mass
m*, the critical temperature Tc, the zero temperature
superconducting gap 5, the Pauli spin susceptibility
6P) are «thermodynamical» properties, meaning that
they are static quantities which can be determined
within the context of the imaginary frequency
Matsubara space. On the other hand, the anomalous
electron-phonon and isotope effects reported in
cuprates by the photoemission spectroscopy give rise
to a deep interest about the role of nonadiabaticity on
frequency-dependent spectroscopic quantities. This
task is however made quite difficult by the need of an
analytical continuation from imaginary to real axis
frequencies, which in the mean-field-like ME theory
results to be relatively simple but which in the
nonadiabatic vertex corrected context becomes very
hard. In this section we summarize thus only non-
adiabatic anomalies which arise from finite band ef-
fects neglecting for the moment the explicit inclusion
of the vertex diagrams. We shall see however that the
photoemission phenomenology presents already at this
level many interesting features. In particular we see
that the following fundamental el-ph properties
[27,70] are no longer valid when EF , �0:
i) the el-ph self-energy �( )� does not renormalize
the electronic dispersion for � much larger than the
phonon energy scale �0;
ii) impurity scattering affects only the imaginary
part of the self-energy but not the real part, and hence
not the electronic dispersion;
iii) different channels of electron scattering (pho-
nons, impurities, ...) sum linearly in the self-energy.
The small Fermi energy effects on the real axis due
to the finite bandwidth can be conveniently dealt with
by means a proper generalization of the Marsig-
lio—Schossmann—Carbotte technique [71], which in
finite bandwidth systems and in the presence of impu-
rity scattering reads [72]:
�
�
( ) ( ) ( ) ( ) ,
( )
i iT i i i
T
n
m
n m m n
m
� � � � 7 � �7 �
� �
�
� � �
�
�
2 2
2 ( , ) ( ) ( )
[ ( ) ( )] ( )
� � 7 � �
� 7 � �7
m m d F
N f
�
�
�
�
��
�
� 8 8
8 8 8
2
( ) ,
( ) ( )[ ( ) ( )]
( )
�
� � �
7 � �7
� 8 8 8 8
8
�� �
�
� � �
��
�
�d F N f2
� �( ) ,�
(41)
(42)
(43)
where N x( ) and f x( ) are the Bose and Fermi distribu-
tion functions, respectively, �2F( )8 is the el-ph
Eliashberg function, � is the impurity scattering rate.
Moreover
� �( ) ( ) [ ]z d F / z�
��
�
� 8 8 82
(z complex number), � � � � � �� �
( , ) Im ( )m mi , and
7 �
� �
( )
( )
,m
F
m m
E
Z
�
�
�
�
�
�
�arctan
7 �
� � � �
� � �
� �
�
� �
�
�
( ) ln
[ ( )] [ ( )]
[ ( )] [
1
2
2 2
2
E Z Z
E Z Z
F
F �
�
�
�
�
�
�
�
�( )]
,
� 2
� �
� �
� �
� �
� � �
� �
� �
�
��
�
�
� �
( )
( )
( )
(
arctan arctan
E Z
Z
E ZF F )
( )
,
� �Z� �
�
��
�
468 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
L. Pietronero and E. Cappelluti
0 0.4 0.8
� 0 /E F
0
–0.04
–0.08
–0.12
–0.16
�
� = 0.7
0 0.4 0.8
�
�0 /E F = 0.7
Fig. 17. Isotope effect on the spin susceptibility as a func-
tion of �0/EF (� � 07. , N I( ) .0 04� ) and as a function of �
(�0 07/EF � . , N I( ) .0 04� ). Solid lines and dashed lines as
in the previous figure.
where Z z z /z( ) ( )�
1 � .
In Fig. 18,a we plot the real and imaginary part of
the self-energy for an el-ph Einstein model with
EF � 4 0� and � � 1 in the presence of impurity scat-
tering. Let us first the nonadiabatic self-energy for
EF � 4 0� with the ME one (EF � 9) in the � � 0 limit
(dashed line). Note that in this latter case the real
part of the self-energy is always negative implying
that the effective electronic band Ek is always less
steep than the bare one k : Ek k� for any energy.
In addition the low-energy part of ��( )� is just
�� ,( )� ��, which gives the well-known renormali-
zation of the electronic dispersion E /k k�
�( )1
close to the Fermi level [27,70], while for � �& 0 the
��( )� goes rapidly to zero implying that no significant
renormalization effect is expected. Note also that the
magnitude of ���( )� is a monotonously increasing
function with � and saturates for � �: 0.
The presence of a Fermi energy of the same order of
the phonon frequencies gives rise to a number of quali-
tative new features. Most important here we would
like to signalize [72]: i) ���( )� is no longer a monoto-
nous function of �, but when � becomes roughly
�� EF the imaginary part of the self-energy starts to
decrease in modulus and it goes quite rapidly to zero.
This is easily understandable in small Fermi energy
systems if one considers that for � && EF there are no
electronic states into which an electron with energy �
could decay within an energy window , �0. Another
interesting feature is indeed the large positive hump of
the real part of the self-energy which occurs in corre-
spondence of the drop of the imaginary part and it
scales with EF . In particular we note that, in contrast
with the case EF � 9, for finite EF the real part of the
self-energy ��( )� becomes positive in a large range of
energy for � �� 2 0. Finally, by looking at the effect
of the impurity scattering, we note that the presence
of two sources (impurities and phonons) of scattering
does not sum linearly neither in ��( )� neither in
���( )� . These anomalous features have an important
impact on the renormalized electronic dispersion ob-
tained by E Ek k k
� � � ( ) 0 which corresponds in
ARPES measurements to the dispersion inferred by
the momentum distribution curves (MDC). As shown
in Fig. 18,b the positive part of ��( )� implies an
«anti-renormalization» of the electron band, namely
Ek k& . This new feature extends up to an energy
scale which does not depend on �0 but only on EF ,
while its magnitude depends on el-ph parameters as �
or EF itself. In such a situation the high-energy part
Ek & �0 of the experimental electronic dispersion
[16,73] does not represent anymore the bare band k
but it expected to show a steeper behavior than k . In
addition the dependence of the electronic dispersion
on the impurity scattering shows that the high-energy
part of Ek is still highly dependent on the microscopic
details not only of the electron-phonon scattering but
also of any other source of electronic scattering, as for
instance here impurities. This consideration could ac-
count for the strong dependence of the high-energy
part of the ARPES data on the hole doping � in
Nonadiabatic breakdown and pairing in high-Tc compounds
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 469
–2
–1
0
1
2
�
‘(
)
�
0 2 4 6 8 10
�
–3
–2
–1
0
�
''(
)
�
a
–4 –3 –2 –1 0
�
k
–5
–4
–3
–2
–1
0
E
k
b
Fig. 18. Panel a: Real and imaginary part of
for a Ein-
stein phonon mode with � � 1 and EF � 4 0� in the pres-
ence of impurity scattering. Solid lines corresponds to (up-
per panel: from bottom to the top; lower panel: from top
to the bottom): � �/ 0 0 02 04 10� , . , . , , .� where � is the im-
purity scattering rate. Energy quantities are expressed in
units of �0. The dashed line in the upper panel is the real
part of the self-energy in the adiabatic infinite bandwidth
limit EF � �. Panel b: renormalized electron dispersion
corresponding (from left to the right) to panel a. The
dashed line represents the adiabatic limit.
cuprates, where the electron-electron interaction is ex-
pected to be strongly dependent on � [73].
4. Nonadiabatic superconductivity in fullerides
and cuprates
In the previous section we have introduced the gen-
eral equations of the superconducting and normal
state theories of nonadiabatic systems. A number of
different properties have been explicitly evaluated in
simple models in order to point out the anomalous
nonadiabatic features arising from the opening of new
electron-phonon interaction channels related to the
vertex diagrams. While the nonadiabatic theory above
discussed would apply in full generality in a wide va-
riety of nonadiabatic systems, specific features of dif-
ferent materials would depend of course on specific
microscopical and materials-science details. In the
course of our discussion we have already linked the
predictions of the nonadiabatic theory with some un-
conventional experimental findings in cuprates and
fullerides. In this last part we would like thus to ad-
dress in more details the specific nonadiabatic super-
conducting properties of these compounds.
In regards with the nonadiabatic features,
fullerides represent a very important test since, due to
the extremely high frequencies of the intra-molecular
phonon spectrum and to the weak inter-molecular
electron hopping [74–76], the nonadiabatic ratio
�ph/EF is significantly large, �ph/EF � 0.4–0.8
[74,75]. On the other hand, in cuprates the two-di-
mensional character of the electronic band and the
closeness of the Fermi level to a logarithmic Van Hove
singularity [77] permit to investigate additional non-
adiabatic effects triggered by the flatness of the elec-
tronic bands near the Van Hove saddle point. In both
these materials the large local Hubbard repulsion U
compared to the small Fermi energy EF gives rise to
strong correlation effects.
4.1. Correlation effects on the electron-phonon
scattering
The interplay between the electronic correlation
and the electron-phonon interaction plays a funda-
mental role within the context of the nonadiabatic
theory of the electron-phonon interaction. To better
understand this issue we remind that the nonadiabatic
effects, especially on the superconducting pairing, de-
pend significantly on the sign of the vertex processes.
In particular we have shown that a predominance of
forward scattering with small momentum | |q would
lead to an enhancement of the effective electron-
phonon coupling by selecting positive regions of the
vertex function [37–39]. From the physical point of
view this situation has been argued in literature to be
naturally realized in systems with a strong degree of
electronic correlation [52–57].
The effects of strong electronic correlations on the
one-particle properties has been already studied in ex-
tensive way in the literature [78–81]. Common wis-
dom describes the evolution from a free-like system to
a correlated one in term of coherent and incoherent
parts of the Green’s functionG( , )k � . In the context of
a Fermi liquid picture the coherent part of G can be
written as [78–80,82]:
G
Z
Zcoh ( , )
( )
,k
k
�
�
�
(44)
where the reduced quasi-particle weight Z which
renormalizes also the quasi-particle dispersion is due
to the electronic correlation and it depends on the mi-
croscopic parameters of the system. In the context of
Hubbard model for instance Z Z U n� ( , ), where U is
the on-site Hubbard repulsion and n is the electron
density per site. One-particle correlation effects can
be thus parametrized in terms of Z where Z , 1 repre-
sents a weakly correlated system and Z � 0 a strong
correlation regime. For Z � 0 a metal-insulator transi-
tion is expected where quasi-particle weight vanishes
[78–82].
Concerning the interplay between the elec-
tron-electron correlation and the electron-phonon in-
teraction a crucial interest is paid to the two-particle
response functions. In particular, since phonons are
directly coupled to the electronic charge, a primary
role is played by the charge density response. Work in
this direction has been mainly based on analytic tools,
as slave-boson techniques [52,57,83–86] or Hubbard
X-operator formalism [53–55], recently supported by
quantum Monte Carlo calculations [56]. Object of in-
terest has been the effect of the electronic correlation
on the phonon mediated electron-electron interaction
� �( , )q q or, equivalently, on the electron-phonon ma-
trix element g( )q which, in the absence of correlation,
can be assumed to be shapeless g g0( )q � . The two
quantities are simply connected by � �( , )q q �
� | ( )| ( )g D qq 2 � , where D q( )� is the phonon propa-
gator.
The evolution of | ( )|g q 2 on the degree of electronic
correlation has been widely studied. In Refs. 52–57 it
was shown that a prominent peak arises for small | |q
scattering by increasing the electronic correlation
while large | |q momenta are strongly suppressed. In
the strong correlation regime this peaked structure in
| |q can be in good approximation described by a
Lorentzian:
470 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
L. Pietronero and E. Cappelluti
| ( )|
| |
,g gq
q
2 2
2 2
1
�
�;
(45)
where ; �1 represents the inverse correlation length
which can be identified with the cut-off parameter qc
introduced in Sec. 2. The small momenta selection be-
comes more and more pronounced by approaching
the metal-insulator transition and for Z � 0, qc � 0
[52–57].
The concept of correlation length can be useful to
get a physical understanding of the small | |q selection.
Let us consider an electron on the site i in a un-
correlated system. In good approximation it interacts
with the other electrons only through their mean den-
sity. The space positions of the single electrons are
thus independent each others (Fig. 19,a) and any spa-
tial charge modulation with wave vector q probes the
same equal electronic response.
Things are different in correlated systems. In this
case indeed the position of one electron at the site i is
correlated with the position of the other electrons
within a radius ; (Fig. 19,b) [79]. The internal dy-
namics within a size ; is thus frozen out and charge
modulations with wavelength l � ; are prevented. In
the momentum space this is reflected in a predomi-
nance of small momenta | |q � �; 1 and in a suppression
of the large ones | |q & �; 1.
We can now understand the empirical relation be-
tween the high critical temperature superconductivity
and the strong electronic correlation on the basis of
the nonadiabatic scenario. Nonadiabatic effects and a
strong electronic correlation are natural by-products
of small Fermi energy systems when EF becomes com-
parable with both the phonon and the Hubbard energy
scales. The first feature (nonadiabaticity) gives rise to
new interaction channels which can enhance or sup-
press the electron-phonon coupling depending on mi-
croscopic details [37–39]; the second one (electronic
correlation, small | |q momenta) will select the positive
part of the vertex processes and as a consequence it
will favor the resulting electron-phonon coupling en-
hancement due to the nonadiabatic effects. According
this picture strongly correlated systems appear as na-
tural good candidate for high-Tc superconductivity
within the context of the nonadiabatic theory [39].
The constructive interplay between nonadiabatic elec-
tron-phonon interaction and electronic correlation,
and the role of the modulation of the small-q scatter-
ing as function of the correlation degree, are pointed
out in the most remarkable way in the fullerene com-
pounds and in cuprates.
4.2. Fullerenes
Superconductivity in fullerenes is commonly asso-
ciated with A C3 60 compounds [74,75]. Pristine C60 is
a band insulator where the highest molecular orbitals
hu are completely filled [74,75]. Chemical intercala-
tion with alkali atoms, which are completely ionized,
provides additional charges and permits to dope these
systems. Band theory would thus predict a metallic
character up to n � 6 electrons for buckyball when the
lowest unoccupied molecular orbitals t u1 are expected
to be completely filled. The actual phase diagram vs. n
is however quite more complex, and a narrow metallic
regime is found only close to n � 3 in A C3 60 com-
pounds [87]. Present understanding of this anomalous
phase diagram is not at all exhaustive and certainly it
needs the proper inclusion of the electronic correlation
and of anomalous lattice features. It is interesting to
note however that superconductivity appears only in
the metallic regime. This observation suggests that the
metallic character is a fundamental requirement for
superconductivity in these systems.
In spite of many characteristic properties of C60
compounds which make them unlikely candidates for
high-Tc superconductivity in the Migdal—Eliashberg
framework (strong electron-electron repulsion, signi-
ficant electronic correlation, low-carrier density), su-
perconductivity in A3C60 has been often regarded as
conventional [88]. The high critical temperatures
(Tc � 33 K in RbCs C2 60, Tc � 40 K in Cs C3 60 under
pressure) are thus attributed to extremely favorable
microscopic values: a particularly strong electron-
phonon coupling � , 1 (LDA calculations estimate
� � 0.5–1) and high-energy phonon frequencies
�ph � 1000–1500 K [89–91]. These features should
thus compensate the relatively small density of states
and the strong electron-electron Coulomb repulsion
�* .� 0 4 [74,92,93]. The Jahn—Teller nature of the
electron-phonon interaction is suggested in addition to
further reduce the effect of the Coulomb repulsion
[94]. In this situation the critical temperatures of the
Nonadiabatic breakdown and pairing in high-Tc compounds
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 471
e
electron
– e
ba
r
n(r)
r
n(r)
–
electron
correlateduncorrelated
�
Fig. 19. Schematic picture of uncorrelated (a) and corre-
lated (b) electrons. Correlated systems are characterized
by the correlation hole (depicted in panel b) which sur-
rounds each electron and prevent charge fluctuations with
wavelength less than
.
C60 compounds are close to maximum values theoreti-
cally achievable.
On a microscopic ground the standard Mig-
dal—Eliashberg theory is however intrinsically incon-
sistent with respect to the adiabatic problem. C60
compounds are indeed characterized by a set of very
narrow bands with typical EF � 0 25. eV, for both
electron and hole doping, whereas phonon mode ener-
gies range up to �ph � 0 2. eV [95,96]. The breakdown
of the nonadiabatic hypothesis is shown in the most re-
markable way in Fig. 20 where the «adiabatic phase
diagram» (�ph/EF vs. �) of fullerenes, obtained by
different numerical calculations, is drawn. In this sit-
uation the nonadiabatic theory outlined in the previ-
ous section is the unavoidably starting point of any
realistic description of superconductivity in these ma-
terials. In addition we can expect that the interplay
between the strong electronic correlation and the
nonadiabatic electron-phonon coupling would en-
hance the superconducting pairing. A significant indi-
cation about the relevance of a nonadiabatic effects is
given by the observed reduction of Tc upon induced
disorder [99]. This feature, as we have previously
seen, can be considered one of the hallmarks of a
nonadiabatic pairing in s-wave superconductors [62].
To illustrate the role of the opening of nonadia-
batic channels in fullerenes, the concrete example of
Rb3C60 can be useful since for this compound best
experimental data are available. Recent measure-
ments have indeed determined with the highest degree
of accuracy the carbon isotope coefficient on Tc,
�Tc
� 0 21. [100], which, together with the large criti-
cal temperature Tc � 30 K, can be used of basis of
analysis to test both Migdal—Eliashberg and the
nonadiabatic theories. In Fig. 21 we show the numeri-
cal solutions of both the adiabatic Migdal—Eliash-
berg and the nonadiabatic theory constrained to repro-
duce the experimental values Tc � 30 K and � � 0 21.
with an Einstein phonon �ph. For given values of �ph
lying in the physical range of the fullerene phonon
spectrum �ph � 2300 K a extremely large electron-
phonon coupling � � 1–4 is required in ME theory
(filled squares), in contrast with local density approx-
imation results which find � � 1. On the other hand,
the same experimental data are fitted in the non-
adiabatic theory (open triangles) with much more rea-
sonable values of � in good agreement with the theo-
retical calculations [95].
We would like to stress that, within the context of
the nonadiabatic theory of superconductivity, the
high values of the critical temperatures in fullerides
are not related to some particularly strong elec-
tron-phonon coupling rather more to the onset of
higher order diagrams scaling with �ph/EF . This per-
spective has interesting consequences in regards with
the optimization of the superconducting properties in
these and in other nonadiabatic superconductors. In
particular, since the electron-phonon coupling is ex-
pected to be in the weak-intermediate regime
� , 0.5–1, the fullerenes are likely to be far from lat-
tice instabilities, which are mostly due to the close-
ness of the Mott—Hubbard metal-insulator transi-
472 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
L. Pietronero and E. Cappelluti
0.8
0.6
0.4
0.2
0 1 2
KC8 Al Sn Nb Pb Hg
Nb Sn3
�ph F/E
�
A C3 60
Fig. 20. Phase diagram of conventional superconductors
compared with the fullerene compounds in the space de-
fined by the electron-phonon coupling � and the adiabatic
parameter �ph/EF. Data for the A C3 60 family where
taken from DFT, tight-binding, and ab initio calculations
[89–91,97,98] by using standard values for the density of
states N EF( ) = 10 states/(eV-spin-C60) and for the Fermi
energy EF � 025. eV.
0.5 1.0 1.5 2.0 2.5 3.0 3.5
�
0
500
1000
1500
2000
�
p
h
[K
]
ME theory
nonadiabatic theory
0
0.2
0.4
0.6
�
�
Fig. 21. Phonon frequency �ph (lower panel) and Cou-
lomb pseudopotential �* (upper panel) as functions of the
electron-phonon coupling �. Both the ME (filled squares)
and the nonadiabatic (open triangles) equations have been
solved in order to reproduce Tc � 30 K and � � 021. .
tion, so that Tc could reasonably be further increased
in fullerides upon increasing the electron-phonon coup-
ling itself.
A widely used way to control the electron-phonon
coupling in fullerides is by tuning the lattice spacing
a. This is usually done in the A C3 60 by varying the al-
kali atoms A with different ionic sizes [74]. Since the
phonon and electron-phonon properties are mainly
ruled by intra-molecular properties, the main effect of
the lattice spacing is of tuning electronic quantities
like the density of states N( )0 and the Fermi energy
EF . Larger lattice parameters a correspond thus to
high density of states and to higher Tc’s. Another con-
venient way to obtain large lattice spacing in
fullerides is by means of ammonia intercalation.
Although at the present ammonia intercalation in
( )NH3 xA C3 60 (x � 1) is usually found to suppress su-
perconductivity, it is important to note that the reduc-
tion of Tc is always accompanied by the reduction of
the symmetry of the system, global (long-range
antiferromagnetism, lattice distorsion) or local one
(crystal field splitting) [101]. According our view
such reductions of symmetry are more likely to be re-
lated to the electron-electron correlations than to the
electron-phonon coupling. In this perspective, we sug-
gest that the electron-phonon coupling and thus the
critical temperature could be significantly further in-
creased in ammonia intercalated compounds once
these reductions of symmetry are prevented. The high
critical temperature Tc � 30 K of (NH A C3 4 3 60)
[102], where the full electron and lattice symmetries
of the A C3 60 are restored, seems to confirm this pic-
ture.
4.3. Copper oxides
In comparison with the C60 compounds the cuprate
family presents a much more complex phenomenology
as it is testified by the richness of anomalous features
in the T vs. � phase diagram, schematically depicted in
Fig. 22. Undoubtedly the revealing of different phases
in cuprates is made easier than in fullerenes by the fact
that the electronic filling in copper oxides can be
tuned in a continuous way by stoichiometric doping,
whereas C60 compounds become rapidly insulators as
soon as the electron concentration per buckyball
moves away from x � 3 (half-filling). In addition, al-
though correlation effects are certainly important in
fullerenes, the A C3 60 compounds lie on the metallic
side of the Brinkmann—Rice transition while cuprates
at half-filling are Mott insulator with long range
antiferromagnetic order. The appearing of a variety of
exotic phases close at half-filling is not surprising
since in this regime the quasi-particle kinetic energy
of the electrons is highly suppressed so that it can be
easily overwhelmed by different long-range or short-
range orderings. At the same time, the smallness of the
kinetic energy scale, parametrized by the Fermi energy
EF , gives rise unavoidably also to nonadiabatic ef-
fects when compared with the phonon energy scale.
As a general rule, the «bell-shape» profile of the
superconducting phase as function of the hole doping
is commonly regarded in a twofold way from the scien-
tific community [103]. According the first point of
view, the onset of some long- or short-range ordering
(AF fluctuations, stripes, spin glasses, pseudogap,
CDW quantum critical point) is the active principle
for the high-Tc superconductivity [104–114]. Along
this perspective the disappearing of these features and
the restoring of normal metal properties in the
overdoping region leads to a reduction of Tc. The
above scenario is reversed according the second point
of view which regards the suppression of Tc in the
underdoping regime as arising from the competition of
the superconducting phase with other different kinds
of ordering [115–118]. In this scenario more emphasis
is thus paid to the overdoped region where other ac-
tors detrimental for the superconductivity are absent
and it should be easier to identify the mechanism un-
derlying the superconducting pairing.
The nonadiabatic theory of superconductivity per-
mits to understand in a very natural way the bell-
shape of the superconducting phase diagram [119].
The active principle in this context is the onset of the
Nonadiabatic breakdown and pairing in high-Tc compounds
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 473
T
overdopedunderdoped
spin glass
antiferromagnetic
spin fluctuations
stripes
(static)
anomalous metal
pseudogap
T
(dynamic)
stripes
optimally doped
Fermi liquid
nearly
( T , a > 1)
a
( T , a < 1)
a
�
AF
SC
( T)� �
Fig. 22. Schematic phase diagram of the copper oxides
compounds in the temperature vs. doping space.
nonadiabatic channels of interaction. A crucial tuning
role however is also played by the electronic correla-
tion which promotes small q scattering and selects the
positive (attractive) parts of the vertex diagrams.
Moving from the overdoped to the underdoped region
we expect thus the electronic correlation to be higher,
the small q selection more effective and the effective
superconducting pairing stronger. This trend, which is
essentially based on the doping dependence of the par-
ticle-particle Cooper interaction, has however to com-
pete with the reduction of the one-particle spectral
weight, which roughly scales with � in the low doping
regime [78,79,82]. The phase diagram resulting from
the competition of these two kinds of effects can
be conveniently discussed in terms of the linearized
superconducting kernel in the nonadiabatic regime
[119], Eq. (37), which can be rewritten in a simplified
way as:
� � � �
�
n c
m
m
m
F
m
T Z Z P Z C
W
E
W
� �
!
""
#
$
%%[ ] ,1 2 arctan
(46)
where the prefactor Z in front of the superconducting
kernel and of the vertex and cross diagrams arises
from the quasi-particle spectral weight reduction ac-
cording Eq. (44), and where we neglect for sake of
simplicity the frequency dependence of the nonadia-
batic contributions P, C. In this way, we can roughly
see the total electron-phonon coupling as the product
of two terms: an effective electron-phonon coupling
of ME theory renormalized by the electronic correla-
tion, �ME , and the enhancement due to nonadiabatic
vertex and cross (vc) diagrams �vc:
� � �
� �
� � �
eff �
�
�
ME vc
ME
vc
c c
Z
Z P Q Z C Q
,
,
( ) ( ) .1 2
The schematic behavior of these quantities as a func-
tion of the hole doping � is shown in the upper panel
of Fig. 23 [119]. The physics behind the �-dependence
of �ME can be easily related to the loss of spectral
weight approaching the metal-insulator transition for
� � 0. This effect, which is present also in �vc, is
however in that case competing with the enhance-
ment of the effective coupling due to P Qc( ) and
C Qc( ) which will be maximum and positive close to
half-filling (where Qc � 0) and negative at high
dopings. The interplay between these two effects will
give rise to a maximum of �vc, and hence of �eff ,
somewhere in the small doping region where the com-
petition between the spectral weight loss and the pos-
itive nonadiabatic effects is stronger (see upper and
middle panels in Fig. 23).
We can now also consider the effect of the residual
Morel—Anderson-like repulsion �; first of all, we ob-
serve that the reduction of spectral weight will lead to
an effective repulsion � �eff � Z . Superconductivity
will be possible only when the net electron-phonon at-
traction overcomes the repulsion term: � �eff eff
& 0
(see lower panel of Fig. 23) [119]. The resulting total
coupling is expected to exhibit a bell-shape which is
mostly due to the �-dependence of the nonadiabatic
factor �vc. It is interesting to note two things. First, in
474 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
L. Pietronero and E. Cappelluti
SC
�0
�vc = 1
�vc
ME
�
SC
eff�
eff�
eff
�
eff
� –
Fig. 23. Graphical sketch of the different contributions to
the effective superconducting coupling. Top panel: the
coupling function �ME is mainly determined by the coher-
ent spectral weight, and it exhibits a monotonous growing
behavior as a function of doping. The vertex factor �vc
tends to enhance the effective coupling at low doping and
to depress it at high doping. Middle panel: the total ef-
fective electron-phonon coupling � � �eff � ME vc has a maxi-
mum at some finite value of �; when the effective Morel—
Anderson pseudopotential is subtracted, superconductivity
is suppressed at high doping. Lower panel: resulting phase
diagram for superconductivity: superconductivity is only
possible in a finite region of phase space (gray region),
where � �eff eff� is positive.
the extreme case � �ME � eff , where no superconduc-
tivity would be predicted in the whole � range by the
conventional ME theory, we could expect finite Tc in
a small � region, due to purely nonadiabatic effects
� � � �eff eff� &ME vc . Secondly, it is clear that within
the ME framework a net attractive interaction in the
Cooper channel at a certain doping �, which corre-
sponds to � �ME & eff , would imply a superconducting
order also at larger � since the two quantities
� �ME, eff scale in the same way � Z; on the other
hand, in the nonadiabatic theory superconductivity,
Tc is expected to be limited to some maximum value of
doping, due to the negative contribution of the
nonadiabatic diagrams P andC at large � (large Qc’s).
We would like to stress that the above phase dia-
gram was drawn keeping in mind a minimal scenario
where the only actors were the enhancement of the
particle-particle interaction due to the nonadiabatic
terms and the reduction of the one particle spectral
weight. It is understood on the other hand that an ex-
haustive description of the different features of the ex-
perimental phase diagram, including also the normal
state, involves unavoidably the taking into account of
a series of other different factors.
Without entering in details, we would like just to
mention several features of the cuprates which appear
unconventional according the Migdal—Eliashberg
theory but which can be naturally accounted for
within the nonadiabatic context. We have previously
already mentioned the onset of unconventional isotope
effects both on Tc, with isotopic coefficient
�Tc
� 0.2–0.8 either larger or smaller than the BCS
limit, and on the effective electronic mass m*, with
� m* � 0. Both these features are indeed experimen-
tally observed in cuprates [6–10]. Another apparently
puzzling issue in cuprates regards the symmetry of the
superconducting order parameter. While the d-wave
symmetry seems well assessed in hole doped systems
[120], the debate is still open about the symmetry in
electron doped materials, where some indications sug-
gest a transition from d- to s- or to other anomalous
symmetries as function of doping, maybe also in the
hole-doped compounds [121–126]. The s-wave symme-
try is on the other hand well accepted in fullerenes. In
this situation it seems difficult to reconcile the differ-
ent phenomenology of these different systems with a
unique superconducting mechanism. In hole-doped
cuprates the d-wave symmetry is often discussed to
rule out an electron-phonon mechanism. This view is
however based on the assumption of an isotropic elec-
tron-phonon scattering | ( )|g g2 2q � , which is signifi-
cantly violated in correlated systems as discussed in
Sec. 4.1. In this situation, the momentum structure of
the attractive electron-phonon interaction + the elec-
tron-electron repulsion can give rise to different sym-
metry of the superconducting order parameter, from s-
to d-wave, depending on the momentum cut-off | |q c
[55,127,128], so that the electron-phonon interaction
results to be a valid candidate for explaining the ob-
servation of different symmetries within the context a
unique pairing mechanism. As a final remark we
would like to mention the report of the linear behavior
in temperature of the resistivity in cuprates at optimal
doping [129]. This strict linear dependence has also
been discussed as an evidence of a nonphononic scat-
tering. However, specific studies show that the linear
behavior <( )T T, can be also naturally explained
within the context of an electron-phonon interaction
by taking into account the Van Hove singularity ef-
fects [130]. Note that the flatness of the electronic
band associated with the Van Hove singularity is ex-
pected to give rise unavoidably to nonadiabatic effects
[131].
5. Conclusion
The interest about nonadiabatic effects in high-Tc
superconductors arises from the experimental obser-
vation of a small energy scale associate with the elec-
tronic dynamics, characterized by the Fermi energy
EF . In the high-Tc materials the Fermi energy is
comparable with the phonon frequency scale �ph
(�ph � EF). In this situation the adiabatic assump-
tion of Migdal’s theorem (�ph/EF � 0), on the basis
of the conventional picture of electron-phonon inter-
action in metals, breaks down. A novel approach,
which takes explicitly into account the nonadiabatic
effects in this new regime, is thus unavoidably re-
quired.
In the present contribution we have shown how a
nonadiabatic theory of the superconductivity and of
the normal state can naturally account for the anoma-
lous phenomenology of various high-Tc compounds,
where we have focused on fullerides and copper ox-
ides. In these compounds, in particular, the origin of
the high critical temperature stems out from the inter-
play between strong electronic correlation and elec-
tron-phonon coupling in the new nonadiabatic regime
where new channels of electron-phonon scattering,
due to the breakdown of Migdal’s theorem, are opera-
tive. The nonadiabatic theory of superconductivity
provides thus an unifying scenario for different kinds
of «exotic» superconductors. Within this framework
it is also interesting to mention the specific case of
MgB2, where the origin of the nonadiabatic effects
is slightly different. In this compounds, indeed,
band structure calculations predict a Fermi energy
EF � 0.4–0.5 [132–134], to be compared with the
Nonadiabatic breakdown and pairing in high-Tc compounds
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 475
relevant phonon frequencies �ph � 70–80 meV
[133–137]. This simple analysis would thus predict a
nonadiabatic ratio �ph/EF � 0.1–0.2, which would
locate MgB2 in the weakly nonadiabatic regime
[138,139]. A peculiar feature of this system is however
that the electron-phonon coupling is mainly concen-
trated in only one-phonon mode, which is charac-
terized by a very strong deformation potential I
[135–137]. In this context new nonadiabatic effects
are thus triggered by the quantum lattice fluctuations
[140,141] which lead to modifications of the band
structure of the same order of the Fermi energy itself
I u EF) * �2 1 2/ , where ) *u2 1 2/ is the root means
square of the lattice displacements. Work is at the mo-
ment in progress to formalize this new kind of break-
down of the adiabatic assumption. We think that fur-
ther research on this field would provide, for this class
of materials as well as in fullerides and cuprates, new
routes for the optimization of the superconducting
properties of the existing materials and for the search
of new high-Tc compounds based on a nonadiabatic
type of pairing.
Acknowledgments
The authors acknowledge fruitful collaborations on
this subject with C. Grimaldi, S. Str�ssler, P. Bene-
detti, M. Scattoni, P. Paci, M. Botti, L. Boeri, S.
Ciuchi and G.B. Bachelet. We also acknowledge fi-
nancial support from the MIUR projects COFIN03
and FIRB RBAU017S8R.
1. For a review on conventional low-Tc metals see: Super-
conductivity, R.D. Parks (ed.), Dekker, New York
(1969).
2. P.W. Anderson and C.C. Yu, in: Highlights of Con-
densed Matter Theory, Proc. of the Int. School of
Physics «E. Fermi», LXXXIX, F. Bassani and M. Tosi
(eds.), North-Holland, New York (1985).
3. P.B. Allen and B. Mitrovic, in: Solid State Physics,
v. 37, H. Ehrenreich, D. Turnbull, and F. Seitz (eds.),
Academic Press, New York (1982).
4. J.G. Bednorz and K.A. M�ller, Z. Phys. B64, 189
(1986).
5. L.C. Bourne, M.F. Crommie, A. Zettl, H.C. zur Loye,
S.W. Keller, K.J. Leary, A.M. Stacy, K.J. Chang, and
M.L. Cohen, Phys. Rev. Lett. 58, 2337 (1987).
6. M.K. Crawford, W.E. Farneth, E.M. McCarron, R.L.
Harlow, and E.H. Moudden, Science 250, 1390
(1990).
7. J.P. Franck, S. Gygax, S. Soerensen, E. Altshuler, A.
Hnatiw, J. Jang, M.A.-K. Mohamed, M.K. Yu, G.I.
Sproule, J. Chrzanowski, and J.C. Irwin, Physica
C185–189, 1379 (1991).
8. G.M. Zhao, M.B. Hunt, H. Keller, and K.A. M�ller,
Nature 385, 236 (1997).
9. J. Hofer, K. Conder, T. Sasagawa, G.M. Zhao, M.
Willemin, H. Keller, and K. Kishio, Phys. Rev. Lett.
84, 4192 (2000).
10. R. Khasanov, D.G. Eshchenko, H. Luetkens, E.
Morenzoni, T. Prokscha, A. Suter, N. Garifianov, M.
Mali, J. Roos, K. Conder, and H. Keller, Phys. Rev.
Lett. 92, 057602 (2004).
11. C. Grimaldi, E. Cappelluti, and L. Pietronero,
Europhys. Lett. 42, 667 (1998).
12. A.S. Alexandrov, Europhys. Lett. 56, 92 (2001).
13. T. Schneider and H. Keller, Phys. Rev. Lett. 86, 4899
(2001).
14. A. Deppeler and A.J. Millis, Phys. Rev. B65, 224301
(2002).
15. P. Paci, M. Capone, E. Cappelluti, S. Ciuchi, C.
Grimaldi, and L. Pietronero, Phys. Rev. Lett. 94,
036406 (2005).
16. A. Lanzara, P.V. Bogdanov, X.J. Zhou, S.A. Kellar,
D.L. Feng, E.D. Lu, T. Yoshida, H. Eisaki, A. Fuji-
mori, J.-I. Shimoyama, T. Noda, S. Uchida, Z. Hus-
sain, and Z.-X. Shen, Nature 412, 510 (2001).
17. G.-H. Gweon, T. Sasagawa, S.Y. Zhou, J. Graf, H.
Takagi, D.-H. Lee, and A. Lanzara, Nature 430, 187
(2004).
18. A.F. Hebard, M.J. Rosseinsky, R.C. Haddon, D.W.
Murphy, S.H. Glarum, T.T.M. Palstra, A.P. Ramirez,
and A.R. Kortan, Nature 350, 600 (1991).
19. K. Tanigaki, T.W. Ebbesen, S. Saito, J. Mizuki, J.S.
Tsai, Y. Kubo, and S. Kuroshima, Nature 352, 222
(1991).
20. R.M. Fleming, A.P. Ramirez, M.J. Rosseinsky, D.W.
Murphy, R.C. Haddon, S.M. Zahurak, and A.V. Mak-
hija, Nature 352, 787 (1991).
21. K. Holczer, O. Klein, S.-M. Huang, R.B. Kaner, K.-J.
Fu, R.L. Whetten, and F. Diederich, Science 252,
1154 (1991).
22. J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zeni-
tani, and J. Akimitsu, Nature 410, 549 (2001).
23. Y.J. Uemura, L.P. Le, G.M. Luke, B.J. Sternlieb,
W.D. Wu, J.H. Brewer, T.M. Riseman, C.L. Seaman,
M.B. Maple, M. Ishitawa, D.G. Hinks, J.D. Jorgen-
sen, G. Saito, and H. Yamochi, Phys. Rev. Lett. 66,
2665 (1991).
24. Y.J. Uemura, A. Keren, L.P. Le, G.M. Luke, B.J.
Sternlieb, W.D. Wu, J.H. Brewer, R.L. Whetten,
S.M. Huang, S. Lin, R.B. Kaner, F. Diederich, S.
Donovan, G. Gr�ner, and K. Holczer, Nature 352, 605
(1991).
25. A.B. Migdal, Zh. Eksp. Teor. Fiz. 34, 1438 (1958)
[Sov. Phys. JETP 7, 996 (1958)].
26. L. Pietronero and S. Str�ssler, Europhys. Lett. 18, 627
(1992).
27. G. Grimvall, The Electron-Phonon Interaction in Me-
tals, North-Holland, Amsterdam (1981).
28. A.A. Abrikosov, L.P. Gorkov, and I.E. Dzyaloshinski,
Methods of Quantum Field Theory in Statistical Phy-
sics, Dover, New York (1975).
29. G. Rickayzen, Green’s Functions and Condensed Mat-
ter, Academic Press, London (1980).
476 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
L. Pietronero and E. Cappelluti
30. G.M. Eliashberg, Zh. Eksp. Teor. Fiz. 38, 966 (1960)
[Sov. Phys. JETP 11, 696 (1960)].
31. L. Pietronero, Europhys. Lett. 17, 365 (1992).
32. I.G. Lang and Yu.A. Firsov, Zh. Eksp. Teor. Fiz. 42,
1843 (1962) [Sov. Phys. JETP 16, 1301 (1963)].
33. A.S. Alexandrov, Phys. Rev. B38, (1988).
34. A.S. Alexandrov and N.F. Mott, Polarons and Bipo-
larons, World Scientific, Singapore (1995).
35. A.J. Millis, R. Mueller, and B.I. Shraiman, Phys.
Rev. B54, 5389 (1996).
36. S. Ciuchi, F. de Pasquale, S. Fratini, and D. Fein-
berg, Phys. Rev. B56, 4494 (1997).
37. L. Pietronero, S. Str�ssler, and C. Grimaldi, Phys.
Rev. B52, 10516 (1995).
38. C. Grimaldi, L. Pietronero, and S. Str�ssler, Phys.
Rev. B52, 10530 (1995).
39. C. Grimaldi, L. Pietronero, and S. Str�ssler, Phys.
Rev. Lett. 75, 1158 (1995).
40. A.S. Alexandrov and P.P. Edwards, Physica C331,
97 (2000).
41. V.N. Kostur and P.B. Allen, Phys. Rev. B56, 3105
(1997).
42. A.S. Alexandrov, Europhys. Lett. 56, 92 (2001).
43. M. Grabowski and L.J. Sham, Phys. Rev. B29, 6132
(1984).
44. V.N. Kostur and B. Mitrovic, Phys. Rev. B48, 16388
(1993).
45. V.N. Kostur and B. Mitrovic, Phys. Rev. B50, 12774
(1994).
46. E.J. Nicol and J.K. Freericks, Physica C235, 2379
(1994).
47. J.K. Freericks, V. Zlatic, W.K. Chung, and M. Jar-
rell, Phys. Rev. B58, 11613 (1998).
48. P. Miller, J.K. Freericks, and E.J. Nicol, Phys. Rev.
B58, 14498 (1998).
49. H.R. Krishnamurthy, D.M. Newns, P.C. Pattnaik,
C.C. Tsuei, and C.C. Chi, Phys Rev. B49, 3520
(1994).
50. P. Paci, E. Cappelluti, C. Grimaldi, and L. Pietro-
nero, Phys. Rev. B65, 012512 (2002).
51. C. Grimaldi, L. Pietronero, and M. Scattoni, Eur.
Phys. J. B10, 247 (1999).
52. M. Grilli and C. Castellani, Phys. Rev. B50, 16880
(1994).
53. M.L. Kulic and R. Zeyher, Phys. Rev. B49, 4395
(1994).
54. R. Zeyher and M.L. Kulic, Phys. Rev. B53, 2850
(1996).
55. M.L. Kulic, Phys. Rep. 338, 1 (2000).
56. Z.B. Huang, W. Hanke, E. Arrigoni, and D. J.
Scalapino, Phys. Rev. B68, 220507 (2003).
57. E. Cappelluti, B. Cerruti, and L. Pietronero, Phys.
Rev. B69, 161101 (2004).
58. G. Baym and L.P. Kadanoff, Phys. Rev. 124, 287
(1961).
59. L.P. Kadanoff and G. Baym, Quantum Statistical
Mechanics, Benjamin, New York (1962).
60. D.J. Scalapino, in: Superconductivity, D.R. Parks
(ed.), Dekker, New York (1969).
61. J.P. Carbotte, Rev. Mod. Phys. 62, 1027 (1990).
62. M. Scattoni, C. Grimaldi, and L. Pietronero, Euro-
phys. Lett. 47, 588 (1999).
63. E. Cappelluti, C. Grimaldi, and L. Pietronero, Phys.
Rev. B64, 125104 (2001).
64. M. Botti, E. Cappelluti, C. Grimaldi, and L. Pietro-
nero, Phys. Rev. B66, 054532 (2002).
65. P. Benedetti, C. Grimaldi, L. Pietronero, and G.
Varelogiannis, Europhys. Lett. 28, 351 (1994).
66. P.W. Anderson, J. Phys. Chem. Solids 11, 26 (1959).
67. D. Fay and J. Appel, Phys. Rev. B20, 3705 (1979).
68. D. Fay and J. Appel, Phys. Rev. B22, 1461 (1980).
69. C. Grimaldi and L. Pietronero, Europhys. Lett. 47,
681 (1999).
70. S. Engelsberg and J.R. Schrieffer, Phys. Rev. 131,
993 (1963).
71. F. Marsiglio, M. Schossmann, and J.P. Carbotte,
Phys. Rev. B37, 4965 (1988).
72. E. Cappelluti and L. Pietronero, Phys. Rev. B68,
224511 (2003).
73. X.J. Zhou, T. Yoshida, A. Lanzara, P.V. Bogdanov,
S.A. Kellar, K.M. Shen, W.L. Yang, F. Ronning, T.
Sasagawa, T. Kakeshita, T. Noda, H. Eisaki, S. Uchi-
da, C.T. Lin, F. Zhou, J.W. Xiong, W.X. Ti, Z.X.
Zhao, A. Fujimori, Z. Hussain, and Z.-X. Shen, Na-
ture 423, 398 (2003).
74. O. Gunnarsson, Rev. Mod. Phys. 69, 575 (1997).
75. W.E. Pickett, in: Solid State Physics, v. 48, H. Eh-
renreich and F. Spaepen (eds.), Academic Press, New
York (1994).
76. E. Cappelluti and L. Pietronero, Phys. Status Solidi
B242, 133 (2005).
77. For a review see: R.S. Markiewicz, J. Phys. Chem.
Solids 58, 1179 (1997).
78. E. Dagotto, Rev. Mod. Phys. 66, 763 (1994).
79. P. Fulde, Electron Correlations in Molecules and
Solids, Springer Verlag, Heidelberg (1995).
80. Correlated Electron Systems, V.J. Emery (ed.),
World Scientific, Singapore (1993).
81. A. Georges, G. Kotliar, W. Krauth, and M. Rozen-
berg, Rev. Mod. Phys. 88, 13 (1996).
82. M.C. Gutzwiller, Phys. Rev. Lett. 10, 159 (1963).
83. M. Lavagna, Phys. Rev. B41, 142 (1990)
84. J.H. Kim and Z. Tesanovic, Phys. Rev. Lett. 71, 4218
(1993).
85. J.D. Lee, K. Kang, and B.I. Min, Phys. Rev. B51,
3830 (1995).
86. M. Mierzejewski, J. Zielinski, and P. Entel, Phys.
Rev. B57, 590 (1998).
87. T. Yildirim, L. Barbedette, J.E. Fischer, C.L. Lin, J.
Robert, P. Petit, and T.T.M. Palstra, Phys. Rev. Lett.
77, 167 (1996).
88. J. Bernholc, Phys. Today 52, 30 (1999).
89. C.M. Varma, J. Zaanen, and K. Raghavachari, Sci-
ence 254, 989 (1991).
90. M. Schluter, M. Lannoo, M. Needels, G.A. Baraff,
and D. Tom�nek, Phys. Rev. Lett. 68, 526 (1992).
91. M. Schluter, M. Lannoo, M. Needels, G.A. Baraff,
and D. Tom�nek, J. Phys. Chem. Solids 53, 1473
(1992).
Nonadiabatic breakdown and pairing in high-Tc compounds
Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5 477
92. O. Gunnarsson and G. Zwicknagl, Phys. Rev. Lett.
69, 957 (1992).
93. E. Koch, O. Gunnarsson, and R.M. Martin, Phys.
Rev. Lett. 83, 620 (1999).
94. J.E. Han, O. Gunnarsson, and V.H. Crespi, Phys.
Rev. Lett. 90, 167006 (2003).
95. E. Cappelluti, C. Grimaldi, L. Pietronero, and S.
Str�ssler, Phys. Rev. Lett. 85, 4771 (2000).
96. E. Cappelluti, C. Grimaldi, L. Pietronero, S. Str�s-
sler, and G.A. Ummarino, Eur. Phys. J. B21, 383
(2001).
97. J.C. R. Faulhaber, D.Y.K. Ko, and O.R. Briddon,
Phys. Rev. B48, 661 (1993).
98. V.P. Antropov, O. Gunnarsson, and A.I. Liechten-
stein, Phys. Rev. B48, 7651 (1993).
99. S.K. Watson, K. Allen, D.W. Denlinger, and F.
Hellman, Phys. Rev. B55, 3866 (1997).
100. M.S. Fuhrer, K. Cherrey, A. Zettl, M.L. Cohen, and
V.H. Crespi, Phys. Rev. Lett. 83, 404 (1999).
101. For a review see: Y. Iwasa and T. Takenobu, J.
Phys.: Condens. Matter 15, 495 (2003) and references
therein.
102. O. Zhou, R.M. Fleming, D.W. Murphy, M.J. Ros-
seinsky, A.P. Ramirez, R.B. van Dover, and R.C.
Haddon, Nature 362, 433 (1993).
103. M.R. Norman, D. Pines, and C. Kallin,
cond-mat/0507031 (2005).
104. J. Rossat-Mignod, L.P. Regnault, C. Vettier, P.
Bourges, P. Burlet, J.Y. Henry, and G. Lapertot, Phy-
sica C185–189, 86 (1991).
105. V. Barzykin and D. Pines, Phys. Rev. B52, 13585
(1995).
106. P. Monthoux and D. Pines, Phys. Rev. B50, 16015
(1994).
107. Ch. Renner, B. Revaz, J.-Y. Genoud, K. Kadowaki,
and O. Fischer, Phys. Rev. Lett. 80, 149 (1998).
108. V.M. Loktev, R.M. Quick, and S.G. Sharapov,
Phys. Rep. 349, 1 (2001).
109. M. Grilli, R. Raimondi, C. Castellani, C. Di Castro,
and G. Kotliar, Phys. Rev. Lett. 67, 259 (1991).
110. J.M. Tranquada, J.D. Axe, N. Ichiwara, A.R. Moo-
denbaugh, Y. Nakamura, and S. Uchida, Phys. Rev.
Lett. 78, 338 (1997).
111. V.J. Emery and S.A. Kivelson, Physica C209, 597
(1993).
112. V.J. Emery and S.A. Kivelson, Physica C235, 189
(1994).
113. C. Castellani, C. Di Castro, and M. Grilli, Z. Phys.
B103, 137 (1997).
114. R.S. Markiewicz, J. Phys. Chem. Solids 59, 1737
(1998).
115. J.W. Loram, K.A. MIrza, J.R. Cooper, and J.L.
Tallon, J. Phys. Chem. Solids 10–12, 2091 (1998).
116. E. Cappelluti and R. Zeyher, Phys. Rev. B59, 6475
(1999).
117. S. Onoda and M. Imada, J. Phys. Soc. Jpn. Suppl.
B69, 32 (2000).
118. J. Stajic, A. Iyengar, K. Levin, B.R. Boyce, and T.
Lemberger, Phys. Rev. B68, 024520 (2003).
119. L. Boeri, E. Cappelluti, C. Grimaldi, and L. Pietro-
nero, Phys. Rev. B68, 214514 (2003).
120. C.C. Tsuei, J.R. Kirtley, C.C. Chi, L.S. Yu-Jahnes,
A. Gupta, T. Shaw, J.Z. Sun, and M.B. Ketchen,
Phys. Rev. Lett. 73, 593 (1994).
121. G. Blumberg, A. Koitzsch, A. Gozar, B.S. Dennis,
C.A. Kendziora, P. Fournier, and R.L. Greene, Phys.
Rev. Lett. 88, 107002 (2002).
122. J.A. Skinta, M.-S. Kim, T.R. Lemberger, T. Greibe,
and M. Naito, Phys. Rev. Lett. 88, 207005 (2002).
123. A. Biswas, P. Fournier, M.M. Qazilbash, V.N. Smo-
lyaninova, H. Balci, and R.L. Greene, Phys. Rev.
Lett. 88, 207004 (2002).
124. N.-C. Yeh, C.-T. Chen, G. Hammerl, J. Mannhart,
A. Schmehl, C.W. Schneider, R.R. Schulz, S. Tajima,
K. Yoshida, D. Garrigus, and M. Strasik, Phys. Rev.
Lett. 87, 087003 (2001).
125. Y. Dagan and G. Deutscher, Phys. Rev. Lett. 87,
177004 (2001).
126. A. Sharoni, O. Millo, A. Kohen, Y. Dagan, R. Beck,
G. Deutscher, and G. Koren, Phys. Rev. B65, 134526
(2002).
127. A.I. Lichtenstein and M.L. Kulic, Physica C245, 186
(1995).
128. P. Paci, C. Grimaldi, and L. Pietronero, Eur. Phys.
J. B17, 235 (2000).
129. See for instance: P.B. Allen, Z. Fisk, and A. Migli-
ori, in: Physical Properties of High-Temperature Su-
perconductors, D.M. Ginsberg (ed.), World Scientific,
Singapore (1989).
130. E. Cappelluti and L. Pietronero, Europhys. Lett. 36,
619 (1996).
131. E. Cappelluti and L. Pietronero, Phys. Rev. B53,
932 (1996).
132. J. Kortus, I.I. Mazin, K.D. Belashchenko, V.P. Ant-
ropov, and L.L. Boyer, Phys. Rev. Lett. 86, 4656
(2001).
133. J.M. An and W.E. Pickett, Phys. Rev. Lett. 86,
4366 (2001).
134. T. Yildirim, O. G�lseren, J.W. Lynn, C.M. Brown,
T.J. Udovic, Q. Huang, N. Rogado, K.A. Regan, M.A.
Hayward, J.S. Slusky, T. He, M.K. Haas, P. Khali-
fah, K. Inumaru, and R.J. Cava, Phys. Rev. Lett. 87,
037001 (2001).
135. Y. Kong, O.V. Dolgov, O. Jepsen, and O.K. An-
dersen, Phys. Rev. B64, 020501 (2001).
136. A.Y. Liu, I.I. Mazin, and J. Kortus, Phys. Rev.
Lett. 87, 087005 (2001).
137. K.-P. Bohnen, R. Heid, and B. Renker, Phys. Rev.
Lett. 86, 5771 (2001).
138. E. Cappelluti, S. Ciuchi, C. Grimaldi, L. Pietronero,
and S. Str�ssler, Phys. Rev. Lett. 88, 117003 (2002).
139. E. Cappelluti, S. Ciuchi, C. Grimaldi, and L. Pietro-
nero, Phys. Rev. B68, 174509 (2003).
140. L. Boeri, G.B. Bachelet, E. Cappelluti, and L. Piet-
ronero, Phys. Rev. B65, 214501 (2002).
141. L. Boeri, E. Cappelluti, and L. Pietronero, Phys.
Rev. B71, 012501 (2005).
478 Fizika Nizkikh Temperatur, 2006, v. 32, Nos. 4/5
L. Pietronero and E. Cappelluti
|