Doping-dependent superconducting properties of two-dimensional metals with different types of interparticle coupling (Review Article)
The superconducting properties of a two-dimensional metallic system with arbitrary carrier density and both local and various types of the indirect boson-exchange attractive interaction in the cases of s- and d-wave pairing are analyzed and reviewed at T = 0. In particular, the possibility of a c...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2004
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irk-123456789-1194682017-06-08T03:05:46Z Doping-dependent superconducting properties of two-dimensional metals with different types of interparticle coupling (Review Article) Loktev, V.M. Turkowski, V.M. Обзоp The superconducting properties of a two-dimensional metallic system with arbitrary carrier density and both local and various types of the indirect boson-exchange attractive interaction in the cases of s- and d-wave pairing are analyzed and reviewed at T = 0. In particular, the possibility of a crossover from the Bose—Einstein condensation regime to Bardeen—Cooper—Schrieffer-like superconductivity with growing carrier density and changing coupling in the case of different pairing channels is discussed. Gaussian fluctuations of the order parameter are taken into account, and the carrier density dependence of the gap magnitude is studied. The role of the form of the interparticle attractive interaction in the physical behavior of the system is also considered. 2004 Article Doping-dependent superconducting properties of two-dimensional metals with different types of interparticle coupling (Review Article) / V.M. Loktev, V.M. Turkowski // Физика низких температур. — 2004. — Т. 30, № 3. — С. 247-260. — Бібліогр.: 69 назв. — англ. 0132-6414 PACS: 74.20.Rp, 74.25.Dw, 74.62.Dh, 74.72.–h http://dspace.nbuv.gov.ua/handle/123456789/119468 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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Обзоp Обзоp |
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Обзоp Обзоp Loktev, V.M. Turkowski, V.M. Doping-dependent superconducting properties of two-dimensional metals with different types of interparticle coupling (Review Article) Физика низких температур |
| description |
The superconducting properties of a two-dimensional metallic system with arbitrary carrier
density and both local and various types of the indirect boson-exchange attractive interaction in
the cases of s- and d-wave pairing are analyzed and reviewed at T = 0. In particular, the possibility
of a crossover from the Bose—Einstein condensation regime to Bardeen—Cooper—Schrieffer-like
superconductivity with growing carrier density and changing coupling in the case of different pairing
channels is discussed. Gaussian fluctuations of the order parameter are taken into account, and
the carrier density dependence of the gap magnitude is studied. The role of the form of the
interparticle attractive interaction in the physical behavior of the system is also considered. |
| format |
Article |
| author |
Loktev, V.M. Turkowski, V.M. |
| author_facet |
Loktev, V.M. Turkowski, V.M. |
| author_sort |
Loktev, V.M. |
| title |
Doping-dependent superconducting properties of two-dimensional metals with different types of interparticle coupling (Review Article) |
| title_short |
Doping-dependent superconducting properties of two-dimensional metals with different types of interparticle coupling (Review Article) |
| title_full |
Doping-dependent superconducting properties of two-dimensional metals with different types of interparticle coupling (Review Article) |
| title_fullStr |
Doping-dependent superconducting properties of two-dimensional metals with different types of interparticle coupling (Review Article) |
| title_full_unstemmed |
Doping-dependent superconducting properties of two-dimensional metals with different types of interparticle coupling (Review Article) |
| title_sort |
doping-dependent superconducting properties of two-dimensional metals with different types of interparticle coupling (review article) |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2004 |
| topic_facet |
Обзоp |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119468 |
| citation_txt |
Doping-dependent superconducting properties of two-dimensional metals with different types of interparticle coupling (Review Article) / V.M. Loktev, V.M. Turkowski // Физика низких температур. — 2004. — Т. 30, № 3. — С. 247-260. — Бібліогр.: 69 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT loktevvm dopingdependentsuperconductingpropertiesoftwodimensionalmetalswithdifferenttypesofinterparticlecouplingreviewarticle AT turkowskivm dopingdependentsuperconductingpropertiesoftwodimensionalmetalswithdifferenttypesofinterparticlecouplingreviewarticle |
| first_indexed |
2025-07-08T15:55:47Z |
| last_indexed |
2025-07-08T15:55:47Z |
| _version_ |
1837094821476433920 |
| fulltext |
Fizika Nizkikh Temperatur, 2004, v. 30, No. 3, p. 247–260
Doping-dependent superconducting properties of
two-dimensional metals with different types
of interparticle coupling
(Review Article)
Vadim M. Loktev
Bogolyubov Institute for Theoretical Physics, Metrologichna Str. 14-b, Kiev, 03143 Ukraine
E-mail: vloktev@bitp.kiev.ua
Vladimir M. Turkowski*
CFIF, Instituto Superior Tecnico, Av. Rovisto Pais, 1049-001 Lisbon, Portugal
E-mail: turk@physics.georgetown.edu
Received June 27, 2003, revised September 30, 2003
The superconducting properties of a two-dimensional metallic system with arbitrary carrier
density and both local and various types of the indirect boson-exchange attractive interaction in
the cases of s- and d-wave pairing are analyzed and reviewed at T = 0. In particular, the possibility
of a crossover from the Bose—Einstein condensation regime to Bardeen—Cooper—Schrieffer-like
superconductivity with growing carrier density and changing coupling in the case of different pair-
ing channels is discussed. Gaussian fluctuations of the order parameter are taken into account, and
the carrier density dependence of the gap magnitude is studied. The role of the form of the
interparticle attractive interaction in the physical behavior of the system is also considered.
PACS: 74.20.Rp, 74.25.Dw, 74.62.Dh, 74.72.–h
1. Introduction
The doping dependence of the various properties of
superconductors (including high-temperature ones
(called HTSCs below) raises many theoretical ques-
tions. In particular, the question of the consistent de-
scription of such a dependence, which has a long his-
tory. Probably the first attempt to solve this problem
self-consistently was made by Eagles in Ref. 1, in
which the author tried to employ for HTSCs his re-
sults for the description of the superconducting pro-
perties of Zr-doped SrTiO3 [2] (see also Ref. 3, where
some experimental data were interpreted as the obser-
vation of the Bose—Einstein condensation (BEC) of
electronic pairs in this compound at low itinerant car-
rier densities, and Refs. 4, 5, where it was theoreti-
cally demonstrated that there were reasons for such an
interpretation). More exactly, the author studied the
dependence of the superconducting gap at T = 0 and of
the mean-field critical temperature on free charge car-
rier densities in two-dimensional (2D) and
three-dimensional (3D) systems with a phonon-like
indirect attraction at low densities. The set of coupled
equations for the gap (gap equation) and the Fermi
energy, or chemical potential (number equation), for
these cases was obtained and analyzed. It was esti-
mated that at such densities the diameter of the elec-
tronic pairs is smaller than the distance between them,
and therefore in fact the superconductivity has to cor-
respond to superfluidity of spatially separated (and in
this sense small, or so-called local) pairs. It is impor-
tant that if one can change (for instance, by doping)
the distance between the particles, the transition from
one regime to another (with strongly overlapped
pairs) becomes in principle possible. Now this phe-
nomenon is well-known as the crossover from BEC to
Bardeen—Cooper—Schrieffer (BCS) superconducti-
vity with changing carrier density.
© Vadim M. Loktev and Vladimir M. Turkowski, 2004
* Present address: Department of Physics, Georgetown University, Washington DC, 20057 USA
In Ref. 1 it was also shown that in 3D systems there
exists a critical value of the attraction below which
there are no local pairs, so that BEC—BCS crossover
is impossible. It is noteworthes that the possibility of
a superfluidity scenario for superconductivity was
proposed by Ogg [6] and Schafroth [7–9] long before
the Eagles’s paper was published.
Later on the problem of the BEC—BCS crossover
was considered by Leggett in [10], where 3D systems
with short-range repulsion and finite-range attraction
were studied. The author has analyzed the dependence
of superconducting properties of the system upon the
dimensionless parameter 1/ akF( ) (a is the scattering
length and kF is the Fermi momentum), which de-
fines the itinerant particle number in the crystal. It
was shown that in the limiting cases the metallic sys-
tem consists of bound «bielectronic molecules» in real
space if 1/ akF( ) � � and of Cooper pairs in k space
when 1/ akF( ) � ��. Nozières and Schmitt-Rink gene-
ralized these results to the case of finite temperatures
and the lattice model with a separable interparticle
nonretarded attractive potential V [11]. They have
demonstrated that the BEC—BCS crossover is smooth
with changing kF and V, when V is larger than the
corresponding critical value for the two-particle
(two-fermion) bound state formation. The foremen-
tioned results were obtained for the case of s-wave
pairing.
The real and unprecedented boom of interest in the
crossover phenomena has started in the second half of
the 1980s after the discovery of copper oxide HTSCs,
materials with an evident and rather unusual depend-
ence of the superconducting properties on the carrier
density. It should be stressed that practically all phys-
ical (and observable) properties of the HTSC com-
pounds are doping dependent, but below we shall con-
centrate on the description of the superconductivity
and superconducting properties only.
Development of a consistent theory of HTSCs still
remains one of the most difficult and one of the most
important problems of modern condensed matter phys-
ics. Due to the complicated (multicomponent) crystal
structure, lowered dimensionality, magnetism, strong
electron correlations, inevitable presence of disorder,
etc. a generally accepted theory of these compounds
has not yet been completed.
During the last decade many models (see, for exam-
ple, the review [12]) which take into account some
(not all) cuprate peculiarities have been proposed to
describe various properties of HTSCs, among them the
doping dependence of the superconducting gap and
the critical temperature. For instance, the 3D
BEC—BCS crossover in the s-wave pairing channel
was studied in Refs. 12–19 for the model with local
(so-called four-fermion, or 4F) attraction, and in
Ref. 20 for the model of a nonlocal although separable
attractive potential. In Ref. 12, additional example
with the on-site repulsion and intersite attraction was
also considered. The role of the order parameter fluc-
tuations at T = 0 was analyzed in Ref. 21 (3D case)
and in Ref. 22 (both 2D and 3D cases). The quasi-2D
model with s-pairing at zero temperature was also
studied in Ref. 23. It can be also noticed that the role
of the number equation in connection with the HTSC
problem was emphasized in Refs. 24.
For a pure 2D system the problem of the crossover
in the isotropic s-wave pairing channel was discussed
at zero T (when the real long-range superconducting
order is certainly possible — the Mermin—Wagner—
Hohenberg—Coleman—Bogolyubov theorem [25]) in
Refs. 12, 15, 18, 20, 26–31 and at finite T (when this
order is characterized by the weak algebraic decay of
correlations) in Refs. 12, 15, 18–20, 28–30, 32, 33.
The metal—superconductor phase boundaries on the
nf —U phase diagram (nf is the number of fermionic
particles, U is the on-site attraction in the «nega-
tive-U» Hubbard model) was investigated in Ref. 34
for T = 0 by means of the dynamical mean-field theory.
The indirect retarded, phonon mediated, interfermion
attraction and the features of the corresponding
BEC—BCS crossover at T = 0 were for the first time
studied in papers [35,36].
The 2D crossover for the anisotropic d-wave pair
symmetry is more important because just such a pair-
ing is usually considered to be observed in HTSCs.
The models which lead to this phenomenon were con-
sidered in Refs. 18, 32, 37 at T = 0. The case of finite
temperatures was studied in Ref. 38, where an effec-
tive (Ginzburg—Landau) potential with carrier-den-
sity-dependent coefficients was also derived. The dop-
ing dependence of the critical temperature Tc in the
strongly correlated electron model with elec-
tron—phonon interaction was studied in Refs. 39, 40.
It was found that the vertex corrections to the elec-
tron—phonon coupling in this model lead to strength-
ening of the d-wave superconductivity. In paper [29]
the cases with different pairing symmetries, s, sext ,
dxy and dx y2 2� , in the 2D system with nearest neigh-
bor (n.n.) and next nearest neighbor (n.n.n.) attrac-
tion in the square lattice were studied at zero and fi-
nite temperatures in order to describe the doping
dependence of the superconducting gap and critical
temperature (see also [41]). The possibility of the
BEC—BCS crossover at zero temperature for s-, d-
and mixed s id� -pairings in the 2D system as a func-
tion of coupling constant was considered in Ref. 42.
The same problem in the quasi-2D Hubbard model
with n.n. attraction at finite T was studied in the pa-
248 Fizika Nizkikh Temperatur, 2004, v. 30, No. 3
Vadim M. Loktev and Vladimir M. Turkowski
per [43], where the doping dependence of the super-
conducting properties was also analyzed.
The 2D model with n.n. attraction and also n.n.
and with n.n.n. hopping at T = 0 was studied in
Ref. 44. It was shown that for some relation between
n.n. and n.n.n. hopping parameters the system proves
always to be in the BEC regime and there is no pairing
at low carrier densities when the coupling is weak (see
also Ref. 37). However, as was stressed in Refs. 31,
45, such a statement cannot be correct. A more general
model with on-site repulsion and n.n. attraction at T =
0 in the s- and d-wave channels was investigated in
Ref. 46, where the role of n.n.n. hoping was also stud-
ied. The s- and d-wave crossover at zero temperature in
the model with a doping-dependent attractive interac-
tion was considered in Ref. 47. It is interesting to note
that the boson—fermion model with electrons and
holes and different kinds of the fermion—boson cou-
pling was proposed in Ref. 48 to unify the Bose—Ein-
stein and BCS collective phenomena. It was shown
that both the regime of superfluidity with local pairs
and the regime of superconductivity with Cooper pairs
take place in different sectors of the model parame-
ters.
Even though the d-wave pairing symmetry is now
considered as the typical and almost imprescriptible
property of the HTSC cuprate compounds [49], there
is experimental evidence that mixed s id� - or even
pure s-wave pairing can exist in some of these materi-
als at certain doping values. Indeed, it was observed
that optimally doped Pr Ce CuO1.855 0.145 4y at low
enough temperatures demonstrates a nodeless gap in-
consistent with pure d-wave symmetry [50], and the
superconducting compound Sr La CuO0.9 0.1 2 reveals
the s-wave pairing near the optimal doping [51]. The
analysis shows [52] that the dominant bulk symmetry
of the order parameter in some cuprates is the ex-
tended (or anisotropic) s-wave one. It was also ob-
served that the crossover from the d-wave in the
underdoped and optimally doped regime to the s id� -
wave pairing in the overdoped regime takes place in
YBa Cu O2 3 7 [53]. A crossover from d-wave to s-wave
pairing with doping near optimal x was found for the
electron-doped Pr Ce CuO2 4x x [54], Pr Ce CuO2 4x x y
and La Ce CuO2 4x x y [55]. However, the s-wave pair-
ing is not always present in underdoped and
overdoped cuprates, but the d-wave order parameter
symmetry in overdoped TlBa Cu O2 3 6+� was found in
Ref. 56.
The results cited above evidently show that the in-
vestigation of the doping dependence of the physical
properties of superconductors is now an important
field in solid state physics. The corresponding ques-
tions have not yet been reviewed in the literature and
demand some generalization. Below an attempt will
be made to analyze the behavior of properties of super-
conducting systems with different interparticle poten-
tials in the s- and d-wave pairing channels as functions
of particle densities and interaction strength. For the
sake of simplicity we restrict here our discussion to the
case T = 0. We briefly survey the main superconduct-
ing properties of different systems. We should kindly
apologize because the results of many authors could
not be included in our paper, since it is very difficult
and in fact impossible to cover all of them in such
wide and rapidly growing field as high-temperature
superconductivity, even within a separate, special and
rather narrow topic.
2. The model and the main equations
The most general and, at the same time, the sim-
plest Hamiltonian, which is usually studied in the the-
ory of superconductivity, can be written as
H t c c c c
V c c c c
� � � �
�
� �
� � � �
nm
n m
n m n
n
n
n
n
n n
, ,
† †
,
† †
0 n nm
n m
n m m n
n
n q
n n nq q
� � � � �� �
� �
�
�
�
V c c c c
g c c X
,
† †
,
†( ) ( )
1
2
2
2 2p q
q
q q qn
n n
n,q
( )
( )
( ) ( ) ( ) ,
m
m X�
�
�
�
�
�
�
� � (1)
where c cn n �� ( , ) is the fermionic field operator
with spin � � �, at the lattice site n and at the time
�, tnm describes the n.n., n.n.n. and other hopping
processes;
is the chemical potential of the system;
the nonretarded interparticle interaction is modeled
by the terms proportional toV0 (the on-site attraction
if V0 0� and repulsion otherwise) and Vnm (n.n. or
n.n.n. interaction). The last two terms in (1) describe
an additional retarded fermion—boson interaction
and the free boson parts of the Hamiltonian, where q
is a boson mode with the coordinate Xn q( ), momen-
tum p qn( ), and frequency �n q( ) and gn q( ) is the
fermion—boson coupling. One can easily pass to the
continuum version of this Hamiltonian by replacing
the n.n. hopping operator tnm by t tnm nm� �� (1
� �( ( )) )a / d2 22 (here a is the intercite distance,
d(= 2, 3) is the dimensionality of the system), intro-
ducing the cutoff radius in the interaction terms, etc.
In the case of the d-dimensional square lattice, the
free fermion dispersion relation in momentum space
has the following form, when the n.n. hopping takes
place
Doping-dependent superconducting properties of two-dimensional metals
Fizika Nizkikh Temperatur, 2004, v. 30, No. 3 249
�
( ) cos ,k � � �
�
�2
1
t ak
j
d
j (2)
where k is a d-dimensional wave vector.
As is well known, it is convenient to calculate the
thermodynamic potential by using the path integral
approach for studying the properties of a quantum
many-particle system. This method is not necessary in
the case of the mean-field solution, but it is extremely
useful (see below) when the fluctuations are studied.
Probably one of the first to apply the path integral
methodics for the description of superconductivity
was Svidzinsky [57].
The partition function of the system is
Z D D S� � �� �† e (3)
with the action
S d H /T� � �
�
�
�
�
�
�
�� �
0
1
�
� � � � � � � �n
n
n
†
,
( ) ( ) ( ) , .
(4)
To study the superconducting properties of the sys-
tem, one should make the Hubbard—Stratonovich
transformation with so-called bilocal fields
�nm ( , )� �1 2 and �nm
† ( , )� �1 2 [57] (it is necessary to
pay attention that in the textbook [57] the functional
integration was developed for the situation when spa-
tially inhomogeneous order parameter of the system
depends upon one space variable only, which excluded
a d-wave symmetry):
exp [ ( ) ( ) ( , ) ( ) ( )]† †� � � � � � � � � �
n m nm m n� � � � �
� �
1 2 1 2 2 1V
D�† exp
| ( , )|
( , )
D d d
V
�
�
�
�
�
�
�
�
�
�
� �
0
1
0
2
1 2
2
1 2
� �
� �
� �
� �
nm
nm
� �
�
� �
� �
�
�
nm n m
n m nm
†
† †
( , ) ( ) ( )
( ) ( ) (
� � � � � �
� � � � �
1 2 1 2
1 2 1 2, ) ,�
!
"
�
�
� (5)
where Vnm ( , )� �1 2 includes the effective interparticle
attraction due to boson coupling after integrations
over the boson fields Xn q( ).
Let us introduce the Nambu spinor
#
#
n
n
n
n n n
( )
( )
( )
,
( ) ( ( ), ( )
†
† †
�
� �
� �
� � � � �
�
�
�
�
�
!
"
"
�
�
�
� � ) .
In these terms the partition function can be for-
mally written as
Z D D D D S� � �# # � � # # � �† † ( , , , )† †
,e
where
S
d d
V
( , , , )
| ( , )|
( , )
† †# # � �
�
�
�
$
� �
0
1
0
2
1 2
2
1 2
� �
� �
� �
� �
nm
nm
%
&
'&
� � (
( � � � �
�
n,m
n
nm nm nm
� � � �
� � �
�
( ) ( )
[ � ( )]
†
1 2 1
2
#
#z t m
nm n m
n m
( )
( , ) ( )� ( )
( )� (
† †
†
�
� � � � �
� � �
2
1 2 1 2
1
�
�
�
�
�
� � # #
# # 2 1 2) ( , ) ,�nm � �
)
*
+
(7)
where � (� � )� � �, � ,
1
2 x y and �� z are the Pauli ma-
trices.
The latter action is diagonal over the fermionic
fields, and therefore the integration over #† and #
can be performed exactly. In this case the partition
function becomes
Z G� ��D D� � -* ( [ ]),exp �
where -[ ]G is the thermodynamic potential, which in
the «leading order» is
� � �
� �
� �
� �
-
�
[ ]
| ( , )|
( , )
,
G d d
V
� �
�
� � �
0
1
0
2
1 2
2
1 2
nm
nmn m
Tr Ln Tr LnG G� ��1
0
1. (8)
The Nambu spinor Green function G satisfies the fol-
lowing equation:
[ ( ) � ( )( )�
� (
� � � � � � �
� �
� � � � � � � �
�
� �
�1 3 1 33nl nl nl
nl
I t z
� 1 3 1 3 3 2
1 2
, ) � ( , )] ( , )
( )
*� � � � � �
� � � �
� �
� �
��nl lm
nm
G
(9)
with anti-periodic boundary conditions for fermions
G Gnm nm( ) ( ).� � � � �1 2 1 2� � � � �
The thermodynamic potential (8) is the most general
form of the superconducting effective action, or Ginz-
burg—Landau potential, with a nonlocal retarded
interparticle interaction. As was already mentioned,
it will be used below to study the fluctuation effects.
The minimization of the thermodynamic potential
with respect to the order parameter and the chemical
potential leads to the following system of the coupled
equations:
�
� � �
-
�nm ( , )
;
1 2
0� (10)
250 Fizika Nizkikh Temperatur, 2004, v. 30, No. 3
Vadim M. Loktev and Vladimir M. Turkowski
�
�
� �
-
Nf , (11)
or
�nm nl lm( , ) ( , ) � ( , );� � � � � � �1 2 1 3 3 2� �V GTr (12)
n Gf z� �Tr � ( ,� � �. /nn (13)
where n N /vf f� is the free fermion density in the
system (v is the volume of the system).
In general, it is very difficult to find the Green’s
function Glm ( , )� �1 2 , and therefore some simplifica-
tions must be applied. In particular, we shall consider
the case of the space and time invariance,
V Vnm n m( , ) ( )� � � �1 2 1 2� �� . In this case the Green’s
function has the following form in momentum space:
G i
i i i
n
n z n n
n
( , )
� ( ) ( , )� ( , )�
*
�
� � � � � � �
�
k
k k k
�
� �
� � �� �� �
2 � �� �2 2( ) | ( , )|
,
k k� i n
and the system of the equations (12), (13) acquires
the following form:
�
�
�
( , )
( )
( , )
( ) | ( , )|
i
d p
V
n
d
d
m
m mm
�
0
�
� � �
k
p
p p
�
� �
(
(
� �
2 2 2 2
( , )
( )
( )
( ) ( )
p k
p k
p k
�
�
� � �
�
�
�
�
�� �g
d k
f b
d
d
m n
2
2
2 220
�
� � � �
;
(14)
n
d p
f
d
d
m mm
� �
� �
�
�
�
�
�
�� �
( )
( )
( ) | ( , )|
.
2
1
2 2 20
�
� � �
p
p p�
(15)
In Eq. (14) V( , )p k is the Fourier transform of the
nonretarded interaction, and the term proportional to
gf b� corresponds to the interparticle attraction
through the boson field; � 0n T n� �( )2 1 is the
Matsubara frequency. The interaction term in (14) is
written in general form; it describes, for example, a
local nonretarded interaction, when
V( )p k� � const, gf b� � 0; a nonlocal (nonretarded)
interaction, when V( )p k� 1 const, but gf b� � 0; a
local retarded interaction, when V( )p k� � const,
gf b� 1 0, and �( )p k� � const, etc. The system (14),
(15) will be analyzed in the next Section for various
forms of the interparticle potential V( , )p k and boson
spectrum �(k).
3. The solutions
3.1. The model with local nonretarded attraction
The problem of the crossover from small to large
fermion density in the model with local attraction was
considered in [1,12–19] for the 3D case and in
[1,12,18,26,31] for the 2D case. The corresponding re-
sults can be reduced to the following.
For the simplest case of local nonretarded attrac-
tion, the interaction parameters in equations (14),
(15) have the following form: V V( , )p k � � const,
gf b� � 0. Therefore, the gap in this case is momentum-
and frequency-independent:
� 2( , ) .�n k � � const
The summation in (14), (15) over frequency can be
easily performed, and one gets the standard system of
equations:
1
2
1
2 22 2
2 2
�
�
��
�
�
��
!
"
""�V
d k
T
d
d( ) ( )
tanh
( )
;
0 �
�
k
k
2
2
(16)
n
d k
Tf
d
d
� �
�
��
�
�
��
!
"
""� ( )
( )
( )
tanh
( )
2
1
22 2
2 2
0
�
�
�k
k
k
2
2�
�
�
�
�
�
,
(17)
which at T � 0 have a simple form:
1
2
1
2 2 2
�
�
�V
d kd
d( ) ( )
;
0 � k 2
(18)
n
d k
f
d
d
� �
�
�
�
�
�
�
�� ( )
( )
( )
.
2
1
2 20
�
�
k
k 2
(19)
Since the gap is momentum-independent, only the
isotropic s-wave pairing regime is possible in the
model with the on-site attraction. Equation (19) is
crucial if one wants to investigate the doping depend-
ence of superconducting properties. As a rule, it is not
taken into account in the standard theory of super-
conductivity (or the BCS theory).
The s-wave pairing regime in the case of a quadratic
dispersion law
�
( )k
k
� �
2
2m
and an indirect pairing with boson («Debye») energy
cutoff 3 � � �
( | ( ) | )D k � was considered in the
mean-field approximation for the 2D and 3D cases
and at low carrier densities in Ref. 1 (see also [4,5]).
In fact, in such a situation the integration over mo-
Doping-dependent superconducting properties of two-dimensional metals
Fizika Nizkikh Temperatur, 2004, v. 30, No. 3 251
mentum d k/d d
� ( )20 can be replaced by the integra-
tion over energy 4� ( )º dº, where 4( )º is the density
of states (DOS); it is constant in the case d � 2 and
5 º in the case d � 3.
It is very easy to solve the system (18) and (19) in
the 2D case. When 6F W77 (W is the free fermion
bandwidth), the solution has a simple form:
2 � 2 22Wº ºF
/mV
b Fe � �0 6| | ;
6� ºF b� | |/ ,2
where 6 0
b
/mVW� � �2 4e is the two-fermion bound
state energy. Obviously, the crossover from super-
fluidity to superconductivity with doping occurs in
the 2D case at any coupling constant; then there ex-
ists the value of ºF when
� 0 for any V.
This is not true in the 3D case, when the
BEC—BCS crossover takes place only when the cou-
pling constant is larger than some critical value Vcr .
This distinction follows from the above-mentioned
difference in the DOS in the gap equation. In the 2D
case, when 4( )º � const, the gap equation has the so-
lution 2 � 2W b| |6 at
� 0 and any coupling con-
stant.
For the 3D system the gap equation has the follow-
ing form:
1
2
1
2
2
2 4 2
�
�
�V
k dk
k0 2
(20)
at
� 0. The integral over k on the right-hand side has
the maximal value 2 4 2mW/( )0 at 2 � 0. Therefore,
a simple estimation for Vcr is given by the relation
1 2 4 2� V mW/cr ( )0 , or V / mWcr � 4 220 .
In the case of the momentum cutoff
d kd
d D
( )
( | ( ) | )
20
3 � �
� � �k ,
the approximate solution can be also easily obtained
in the 2D case:
2 2 2� 2| | ( ) ( );6 3 � 3 �b BCS D F BCS F Dº º� � �
6
� �ºF
b| |
,
2
where 2 BCS D
/ mV� �2 2� 0e eff( ) is the BCS expres-
sion for the gap, and the bound state energy in this
case also exists: 6 � 0
b D
mV� � �2 4e eff/( ), where Veff
is proportional to the fermion—boson coupling con-
stant gf b� . The gap is a growing function with dop-
ing in this case, and asymptotically approaches its
maximal value 2 2� BCS when ºF D�� � . In other
words, the dependence of the gap value on the carrier
density has no maximum, which evidently means that
such a dependence with saturation is not identical to
the «increasing—decreasing» dependence of the gap
upon the effective coupling constant Veff .
In the 3D case, the DOS in equations (18) and (19)
can be substituted by the DOS at the Fermi level, and
the solution can be obtained from the 2D result with
substitution mV/ k mV/F( ) ( )2 4 20 0� . It is possible
to estimate the critical value of the coupling constant
when the crossover takes place: V / m Dcr � 2 20 � . In
both cases, with and without momentum cutoff, the
BEC—BCS transition from superfluidity to supercon-
ductivity is smooth and is not a phase transition; the
gap value grows continuously with doping.
3.2. The models with the nonlocal nonretarded
interaction
It is important to study a more realistic case of the
nonlocal attraction in presence of a short-range Cou-
lomb repulsion. In order to study the superconducting
properties of such a model in the channels with differ-
ent pair angular momentum l, it is convenient (see, for
example, Refs. 32, 33, 46), to approximate the inter-
action potential by a separable function:
V w wl l l l( , ) ( ) ( ),k k k k1 2 1 2� �8 (21)
where 8 l is an effective coupling constant, and
w h ll l( ) ( ) ;k k k� cos 9 (22)
h
k/k
k/k
l
l
l /
( )
( )
( )
;k �
� �
1
0
1 21
(23)
k � | |k is the momentum modulus and 9 k is the momen-
tum angle in polar coordinates k k k� k(cos ,sin )9 9 .
Parameters k0 and k1 put the momentum range in the
proper region — the potential is attractive at
r r r0 17 7 and repulsive at r r7 0, where k /r0 015 and
k /r1 115 . The separable form of the interparticle po-
tential (21) is based on the group theory decomposi-
tion of the arbitrary potential by, for example, spher-
ical harmonics, and the restriction to the relevant
terms only, which in our case correspond to the or-
der-parameter symmetry under consideration.
It is easy to see that the interaction potential (21)
has the correct asymptotic behavior at small and large
momenta:V k kl
l l( , )k k1 2 1 25 andV / k kl ( , )k k1 2 1 215 ,
respectively. Since the region of low carrier concentra-
tions, where the crossover can take place, is the most
interesting, the correct behavior of the interaction po-
tential at small momenta should be most important.
These momenta give main contribution to the
integrals in the case of low carrier concentrations (see
equations below). We shall study the s- and d-wave
channels with l � 0 and 2 separately, so we assume that
the parameters 8 l for both channels are independent.
252 Fizika Nizkikh Temperatur, 2004, v. 30, No. 3
Vadim M. Loktev and Vladimir M. Turkowski
In this case the equations for the gap and for the
chemical potential have the following form:
2
2
2
l l
l
l
l
d
V( )
( )
( )
( ) ( )
( , );k
p p
p p
p k� �
�
�8
0 62 2
2 2 2
(24)
n
d
f
l
� �
�
�
�
�
�
�
�
�
�
k k
k k( )
( )
( ) ( )2
1
2 2 20
6
6 2
. (25)
The solution of equation (24) has the following form:
2 2l l lw( ) ( ),( )k k� 0 (26)
where 2 l
( )0 does not depend on the momentum k.
As shown in Refs. 32, 33, the BEC—BCS crossover
from superfluidity to superconductivity is smooth
with doping increasing (see Figs. 1 and 2 below).
However, in the d-wave paring channel there exists a
critical value of the interaction potential, below
which this crossover is impossible. Such a circum-
stance makes the cases of isotropic s- and anisotropic
d-wave pairing essentially different.
The more realistic case in connection to HTSCs was
considered in Ref. 47, where the correlation length r0
was studied at small carrier densities as r a/ nf0 5 .
This dependence for the length of spin-spin correla-
tions at small carrier densities was found, for example,
in La Sr CuO2 4�x x . The magnetic correlation length
decreases with carrier density per cell in this material
as 3 8. Å/ nf [59]. The value of a was considered to be
a / a� 2 00 , where a0 is the simple square lattice con-
stant. The next equality can be used to estimate the
value for a: ( )0/ r N a Nf2 0
2
0
2� cell , where in the left-
hand side the volume of the 2D system is expressed as
a volume (circle of the radius 5 r0) occupied by one
particle, multiplied by the total number of particles
Nf , Ncell is an elementary cell number in the layered
system. The free fermion bandwidthW is related to a0
as W / ma� 02
0
2( ). It should be noted, that the rela-
tion r a/ nf0 5 at a / a� 2 00 is in a good agreement
with the experimental data for La Sr CuO2 4�x x [59],
where the magnetically ordered (orthorhombic) plane
lattice parameters are equal to 5 354. Å and 5 401. Å,
and the corresponding parameter a is � 3 8. Å.
It was shown that the critical value of the coupling
constant exists even in the s-wave pairing channel for
this case (see Figs. 1 and 2).
Let us consider different versions of the previous
model. Namely, the correlation radius r nf0( ) �
� a / nf0 2 ( )0 can be introduced in a model with an
exponential decay of the attraction:
V t g
rb f b
r/r
( , )r � �
�e 0
. (27)
Its Fourier transform has the following form:
D
g r
r
b
f b
( )
( )
.q
q
�
�
� 0
2
0
2 22 10
(28)
In addition, the similar kind of the short-range
fermion—fermion repulsion should be taken into ac-
count. It is easy to see, that in the 2D case this poten-
tial has the following form (see, for example,
Ref. 60):
D
g
q q g q/ kf
f f
TF F
( )
( )
,q �
�
�
2
(29)
where g ef f� � 2 20 , q e m/ / aTF B� �4 42 0 0 is the
Thomas—Fermi momentum, and a / e mB � 1 2( ) �
Doping-dependent superconducting properties of two-dimensional metals
Fizika Nizkikh Temperatur, 2004, v. 30, No. 3 253
0
2.5
5
0.25 0.5
m
8 s
/(
4
0)
ºF
SF SC
0
2.5
5.0
5e-05
m
8 s
/(
4
0)
Fig. 1. Crossover line coupling-carrier density for the
s-pairing channel (solid line). The dotted line represents
the corresponding curve for the case r nf0( ) � const at
r a0 0� . The inset shows the doping dependence of the
crossover value for coupling at very low charge carrier
densities [47]. Here and below all parameters are ex-
pressed in units of the bandwidth W.
0
2.5
5.0
0.25 0.5
m
8
d
/(
4
0)
·1
0
–
4
ºF
SF
SC
Fig. 2. The crossover line coupling-carrier density for the
d-wave case (solid line). The dotted line is the crossover
curve for the case r nf0( ) � const at r a0 0� [47].
� 0 529. Å is the Bohr radius. The function g x( ) is de-
fined as
g x x /x( ) ( ) .� � � �1 1 1 1 23 (30)
This model of the s-wave pairing channel demon-
strates the crossover from superfluidity to supercon-
ductivity at any value of the coupling constant,
contrary to the previous case (see Fig. 3). The do-
ping dependence of the gap and of the chemical po-
tential at different values of the dimensionless in-
teraction parameters 8 0f b f bg mr /� �� 2
0
28( ), 8 f f� �
� �g mr /f f
2
0 4( )0 is presented in Figs. 4 and 5.
As it follows from these figures, the gap decreases
with doping growing at large ºF . This situation is in
a qualitative agreement with the experiments for the
cuprates. One can consider also an interesting situ-
ation of the combined local + nonlocal attraction,
when the local (on-site) attraction will tend the
Cooper pairs to transform into the local pairs.
The «mixed» case with hl ( )k � 1 and 8 8� �s
� 8 9d cos 2 k was considered in Refs. 41, 42. In par-
ticular, it was shown in the paper [41] that the cross-
over from d-wave to s-wave superconductivity with
the intermediate s d� -symmetry takes place with dop-
ing in full agreement with the experiments on some
cuprate materials [53–55].
The d-wave case, when w k kd x y( )k � �cos cos
was considered in Refs. 37, 44, 45. For example, the
effect of the n.n.n. hopping tn n n. . . on the pairing was
254 Fizika Nizkikh Temperatur, 2004, v. 30, No. 3
Vadim M. Loktev and Vladimir M. Turkowski
0
0.5
1.0
1.5
2.0
0.25 0.5
8
f-
b
cr
o
ss
ºF
Fig. 3. The crossover interaction value 8f b�
cross (defined by
the condition
� 0) as the function of ºF at 8f f� � 0.
0
0.05
0.10
0.25 0.5
2
ºF
a 8f-b =0.005
0.010
0.015
0.020
0 0.1 0.2
–0.2
0
0.2
ºF
b
8 f-b=0.005
0.010
0.015
0.020
Fig. 4. The gap (a) and the chemical potential (b) as the
functions of ºF at different 8f b� and 8f f� � 0.
0
0.025
0.050
0.25 0.5
2
ºF
a 8 f-f =0.0000
0.0001
0.0005
0.0010
–0.10
–0.05
0
0.05
0.10
0 0.05 0.1
ºF
b
8 f-f =0.0000
0.0001
0.0005
0.0010
Fig. 5. The gap (a) and the chemical potential (b) as
functions of ºF at different 8f f� and 8f b� � 001. .
studied in Refs. 44, 45. As it was shown in latter pa-
per, the crossover with doping takes place at any value
of tn n n. . ., when the coupling constant is larger than
Vcr .
At last, it was stated in Ref. 29, that the model
with small on-site repulsion and n.n. Vn n. . and n.n.n.
Vn n n. . . attraction with Vn n n. . . 5 60–80 meV and
V /Vn n n n n. . . . . � 1.3–1.5 can describe well the experi-
mental data for the hole-doped oxides.
3.3. The models with retarded interaction
As it is well-known, the retardation effects in the
interaction can play very important role in the super-
conducting properties of the system. For instance, let
us consider a boson propagator with the dispersion
� �� ( )k (cp. (14)):
D
i
( , )
( )
( )
.�
�
� � �
k
k
k
�
� �
2
2 2
(31)
It can be noted that in the general case of phonon
dispersion one has � �( )k k� �0
2 2 2cph . We shall
describe this rather general case below with the
following approximation: D D F( , ) ( , )� �k k� , so the
effective propagator can be written as
D
i
( , ) ,�
�
� � �
k �
� �
0
2
2
0
2
(32)
where �0
2 2 2� c Fphk is substituted by new effective
frequency �0. This approximation corresponds to the
case of the optical phonon attraction (in the case
when �0 01 ). Generally speaking, the gap is fre-
quency-dependent in this model. The problem of the
crossover in the model with frequency dependent gap
was investigated in the Ref. 36.
The set of the equations for the gap and for the
chemical potential in this case is:
�
�
�
( )
( )
( )
( ) | ( )|
�
:
0
:
: � : �
�
� � (
(
� � �
�
�
ig
d kd
i
f b
2
2
2 2 2 2
0
2 k
2
2
0
2( )
;
� : � �� � � i
(33)
2
1
0
2 2
2 2
º
d
F �
�
�
�
�
�
�
��
!
"
""
� �
�
�Re
| ( )|
( | ( )| )�
�
� �
3
� �
�
� �
�
�
�
�
�
�
�
�
��
!
"
""
� �
�
�
�
�
�
� �
3
� �
2 2
2 21
| ( )|
( | ( )| ) .
�
� (34)
Let us consider briefly how this system of the equa-
tions can be analyzed analytically. First of all, it is
possible to show that the approximation � 2 �( )� �
� const in (34) is rather good. Then, this equation re-
sults in
6
� � �º
º
ºF
F
F
b22
2 2
�
| |
,
where the two-particle bound state energy 6b depends
on the coupling parameter in this case (see below).
After the Wick rotation � �� i the gap equation
reads:
�
�
�
( )
( )
( )
( ) ( )
� �
:
0
:
:
:
� (
(
� � �
�
�
g
d kd
/ m
f b
2
0
2
2
3 2 2 2 22 2
1
k ( )
.
� : �� �2
0
2
(35)
In polar coordinates, after integration over the angle
we come to:
�
�
( )
( )
( )
( )
(
�
�
0
:
:
� : �
�
�
�
� �
(
(
�
�
�
�
g
d
k dk
/ m
f b
2
0
2
2 2
0
2
2
2
2k � � �
�
�
: :
;
) ( )
.
2 2 2�
Since �( )� is the even function of �, it can depend
only on �2, and we restrict the integration over : to
the positive values:
�
�
�
( )
( )
( ) ( )
�
8� : :
: : � : �
0
�
� � �
(
( �
�
�0
2
2 2
0
2
0
22
1
2
d
arctan
: :2 2�
�
�
�
�
�
�� ( )
, (36)
where the dimensionless coupling constant
8 0�
�
g m/
f b
2 2( ) is introduced. The asymptotic be-
havior for �( )� is
� �( )| , ( )| .� �
�
� �� ��� 50 2
1
const (37)
As next step we use the following approximation in
the interaction potential [61,62]:
1 1 1
2
0
2 2
0
2 2
0
2( )
( ) ( ).
� : � � �
3 � :
: �
3 : �
� �
�
�
� �
�
�
(38)
Then, the differentiation with respect to � gives:
Doping-dependent superconducting properties of two-dimensional metals
Fizika Nizkikh Temperatur, 2004, v. 30, No. 3 255
�
�
�
< � �
�
(
(
�
��
( )
( )
( )
( )
�
8� :
: �
: :
: : 0
�
0
2
2
0
2 2
2 2
0
1
2d
arctan
: :2 2�
�
�
�
�
�
�� ( )
. (39)
It is evident that �< 7 0, i.e. � � 2max ( )� �0 .
After one more differentiation and introduction of a
new variable x /� : �2
0
2, one gets the following differ-
ential equation
� �
�
<< �
�
< �
� �
(
( �
( ) ( )
( ) [ ( ) ]
x
x
x
x x x x /
2
1 4 1
1
2
2
0
2
8
�
0
arctan
�
�
/
x x /
x0
0
2
0
�
�
�
�
�
�
�
�
[ ( ) ]
( )
�
� (40)
with the boundary conditions
� � �< � � � < �� ��( )| ; [ ( ) ( ) ( )]| ,x x x xx x0 0 1 0 (41)
which follow directly from expressions (36), (38) and
(39).
The analysis of Eq. (40) shows that the approxi-
mate solution for the gap in both weak and strong cou-
pling regimes is (see the paper [36] for details):
2 2( ) ( ),� 3 � �� 0
2 2� where 2 is the parameter, which
depends on the coupling constant and, what is very
important, on the carrier density. In the weak cou-
pling regime the parameter 2 has the following cou-
pling and carrier density dependence:
2 2� � � �2 0 0 0| | ( ) ( ),6 � 3 � 3 �b F BCS Fº º
where 2 BCS
/� �2 0
1� 8e is in fact the BCS expres-
sion for the gap, and the bound (Cooper) state energy
in this case is 6 � 8
b
/� � �2 0
2e . In the strong coupling
regime 6 8b � and 2 BCS /� ( )4 3 08� at large carrier
densities [35,36].
To summarize, the BEC—BCS crossover with car-
rier density and coupling constant changing in this ef-
fective model with retarded interaction is also smooth,
and the gap is not small when the pair frequency is
smaller than the boson frequency. Such an approach
can be used for studying the crossover in the cases of
different symmetries of the order parameter.
It is important to mention that the case considered
here can be easily generalized on the case when one
takes into account the vertex correction to the elec-
tron—phonon interaction. This correction is usually
small when ºF �� �0 (the Migdal theorem). As it was
shown in Ref. 63, this correction is rather small even
when ºF 77 �0. However, in some cases the vertex
correction can lead to strong enough enhancement of
the superconductivity [40,64,65]. It is also necessary
to note, that this correction leads even to the enhance-
ment of the d-wave superconductivity in strongly cor-
related electron system [39,40], despite the phonon
interaction due to the symmetry does not allow the
d-wave pairing in the case when the short range elec-
tron repulsion is not considered.
4. The role of the order parameter fluctuations
The fluctuations of the order parameter in the 2D
and even in the 3D case at T � 0 should be essential.
As it is shown in Ref. 22, the Gaussian fluctuation cor-
rections to the s-wave order parameter is non-negligi-
ble even in the weak-coupling case. On the other
hand, the fluctuations of the order parameter phase
can lead to increasing of the gap.
In this Section we shall consider how simultaneous
order parameter modulus and phase fluctuations in the
model with 4F-attraction result in the strong increas-
ing of the order parameter at small carrier densities
and to small decreasing of the order parameter when
the carrier densities are large.
At zero temperature, as follows from (8), the ther-
modynamic potential of the system with local attrac-
tion has the following form:
-
�
�� � � �
�
�
�
�
�
��v
V
d k| |
( )
[ ( ) | | ( )]
2 2
2
2 2
20
� �k k .
In other words, it depends on the sum of its real and
imaginary parts: | | (Re ) (Im )� � �2 2 2� � . For study-
ing fluctuations of the order parameter, it is conve-
nient to use new real variables:
�( )
( )
( )
x
x
x
�
�
�
��
!
""
=
=
1
2
,
such that � �2
1
2
2
2 2( ) ( ) ( ) | ( )|x x x x� � �= = . Another
possibility is to use the decomposition of the order
parameter to its phase and modulus � 2( ) ( )x x� (
( exp [ ( )]i x3 . However, it leads to some difficulties,
since one needs to keep the order parameter modulus
positive in the functional integration over the fluctu-
ations, therefore we follow another way. The «old»
order parameter variables are connected with new
ones as
� �( ) ( ) ( ), ( ) ( ) ( ).*x x i x x x i x� � � �= = = =1 2 1 2
We assume that the mean-field value of the field � is
chosen as
�
2
0 0
( )x �
�
�
�
!
" .
256 Fizika Nizkikh Temperatur, 2004, v. 30, No. 3
Vadim M. Loktev and Vladimir M. Turkowski
The order parameter � can be written as
� � �
2
( ) ( )
( )
( )
x x
x
x
� � �
��
�
��
!
""0
1
2
�
�=
�=
(42)
in the case when its fluctuations are considered. We
neglect the fluctuations of the carrier density nf ( )r ,
and consider the homogeneous constant value of nf
over the lattice n nf f( )r � � const.
Substitution of (42) into the expression for the
thermodynamic potential gives the following correc-
tion to the thermodynamic potential to the second or-
der in fluctuations [66]:
�
:
0 0
� : : � :- � �� � � �
��
�
� �
d d k
i A i i
2 2
2
2( )
( , ) �( , ) ( , ),k k k
(43)
where �( , ) �( , )A i /V i: > :k k� �1 is the 2 2( matrix
with the susceptibility components
> :
�
0
0
�
jk
j k
i
d
d q
G i
( , )
( )
( ) ( , )�
k
q
� (
( �
��
�
�
� �
�
�
1
2 2
2
1
2
2
Tr
� � �j kG i( , )� ,� �q
where � � :, ,� , � ,/ /2 2, q k q , and j k x y, ,� .
The integration over � can be easily performed:
> :
0 :
� �
11
2
2 2 2
2
2
1
2
1
( , )
( ) ( )
i
d q E E
E E
E
k � �
�
� �
(
( �
�
� � �
� �
� � 2
� �
�
�
�
�
�
�E
;
> : > :
0
� �
:
12 21
2
2 22
1
2
( , ) ( , )
( ) [ (
i i
d q E E
E
k k� � �
� �
�
�� � � � �
� � � � �E E E) ]
;
2
> :
0 :
� �
22
2
2 2 2
2
2
1
2
1
( , )
( ) ( )
i
d q E E
E E
E
k � �
�
� �
(
( �
�
� � �
� �
� � 2
� �
�
�
�
�
�
�E
,
where E, ,� ��2 22 and �
, � , �( ) ( )k q/ / m2 22
is the free fermion dispersion relation. After the inte-
grating out of the fluctuation field �, the correction
to the thermodynamic potential has the following
form:
�
:
0 0
> :- � ��
�
(
��
�
� �
1
2 2 2
12
2 11
d d k
V
i
( )
ln { ( , )}k
( � �
�
�
�{ ( , )} ( , ) ( , ) .
1
22 12 21V
i i i> : > : > :k k k
The factor 5 2 which appears in the measure of the
functional integration over 2 in Z due to taking into
account the symmetry of the thermodynamic potential
with respect to the transformation � �� e i? (see, for
example, the paper [66]) is omitted in the last expres-
sion. This factor can be absorbed in the measure of the
functional integral over 2 in the partition function,
where the functional integration can be actually per-
formed over the variable 22. Let us note that only the
first component ( ( , ))1 11/V i� > : k under the loga-
rithm in �- (see below) will be present if one consid-
ers the particular case of the order parameter phase
fluctuations.
It is useful to diagonalize the matrix �( , )A i: k , in or-
der to find the contributions which come from both
the phase and the modulus fluctuations of the order
parameter. Obviously, the first component will corre-
spond to the phase fluctuations and the second — to
the modulus fluctuations, as follows from the defini-
tions of the field � (see (42)). So, one can easy arrive
at the following representation:
�
:
0 0
> : > :3- 2� � �
��
�
� �
1
2 2 2
1 12
2
d d k
V
i
V
i
( )
ln { ( , )}{ ( , )}k k�
�
�
��
,
where
> : > : > :
> : >
3( , ) [ ( , ) ( , )]
[ ( , )
i i i
i
k k k
k
� � �
� �
1
2
1
4
11 22
11 22( , )] ( , ) ( , )i i i: > : > :k k k2
12 21�
and
> : > : > :
> : >
2( , ) [ ( , ) ( , )]
[ ( , )
i i i
i
k k k
k
� � �
� �
1
2
1
4
11 22
11 22( , )] ( , ) ( , )i i i: > : > :k k k2
12 21�
are the effective contributions to the thermodynamic
potential from the fluctuations of the order parameter
phase and modulus, respectively.
The equations for the gap and for the chemical po-
tential (10) and (11) have the following form in the
case of Gaussian correction to the thermodynamic po-
tential due to the order parameter fluctuations:
2 2
2V
d k
�
�
��
2
2 2 22 2( ) ( )0 � k
�
� �
�
�
�
��
�
� �
1
2 2 2 1
2
2
d d k i /
/V i
i:
0 0
> :
> :
>3
3( )
( , )
( , )
(k
k
2 2 :
> :
, )
( , )
;
k
k
/
/V i
�
�
�
�
�
�
�
2
21
Doping-dependent superconducting properties of two-dimensional metals
Fizika Nizkikh Temperatur, 2004, v. 30, No. 3 257
n
d k
f � �
�
�
�
�
�
�
�
��
2
2 2 22
1
( )
( )
( )0
�
�
k
k 2
�
� �
�
�
�
��
�
� �
d d k i /
/V i
i:
0 0
> :
> :
> :3
32 2 1
2
2( )
( , )
( , )
( ,k
k
2 k
k
)
( , )
.
/
/V i
�
�
�
�
�
�
�
> :1 2
Considering the limit of a weak coupling, the func-
tions >3 and >2 in the denominators can be neglected,
then the integration over : can be performed and after
some trivial manipulations the following simple
enough equation can be obtained (compare with (18)
and (19)):
1
2
1
2 2 2 2
2
2 2 2
2
2
2
2 3V
d k V d q d k
E
�
�
�� � � � �
�( ) ( ) ( ) ( )0 � 0 0
� �
k 2 E�
;
(44)
n
d k
V
d q d k
f � �
�
�
�
�
�
�
�
�
�
�
�
2
2 2 2
2
2
2
2
1
2
( )
( )
( )
( ) (
0
�
�
0
k
k 2
2 2
2
30
� �
)
( )
.
2 � �
� �
�
� E E
(45)
The substitution k k� � in a part of the terms was
made during obtaining equations (44) and (45). It is
interesting to note that in the case of the phase fluctu-
ations, the numerator under the integral in the last
term of the gap equation will be 1
2 � � �� � ��( ) and
the last term in the number equation will be multi-
plied by 1 2/ . It is also a good approximation to put
� ºF , since in the weak coupling regime,
is differ-
ent from ºF only at extremely low carrier densities.
The solution of the equation (44) for the gap parame-
ter as a function of ºF at
� ºF and different values
of the coupling constant is presented in Fig. 6. The es-
timation of the order parameter in the case of phase
fluctuation is also presented. As it was shown in
Ref. 22, in the 2D case the phase fluctuations lead to
an effective increasing of the coupling constant
V V /� �( )1 2 20 . The gap can be calculated from the
standard mean-field BCS equation. The comparison of
different cases shows that the phase fluctuations lead
to the gap increasing, while the total fluctuations lead
to much stronger increasing of the gap at small carrier
densities and to decreasing of the gap when the carrier
density is large. The last result is familiar, but the
first one is very surprising. The dependence of the gap
on coupling at small value of the carrier density is pre-
sented in Fig. 7. These results suggest that higher or-
der corrections in fluctuations should be studied for
better understanding of the behavior of the system.
It should be mentioned that the role of the disorder
due to the dopants in the fluctuations of the
inhomogeneous order parameter was recently studied
258 Fizika Nizkikh Temperatur, 2004, v. 30, No. 3
Vadim M. Loktev and Vladimir M. Turkowski
0
0.05
0.10
0.25 0.5
2
ºF
G=0.3
0
0.1
0.2
0.25 0.5
2
ºF
G=0.5
Fig. 6. The dependence of 2 on ºF for the mean-field so-
lution case (solid line) and for the order parameter fluctu-
ations case (dashed line) at different values of the
dimensionless coupling parameter G mV/� ( )20 . The dot-
ted line is the estimation from [22] for the case of the or-
der parameter phase fluctuations.
0
0.05
0.10
0.15
0.20
0.50
2
G
0.25
Fig. 7. The dependence of 2 on G for mean field solution
case (solid line) and for order parameter fluctuations case
(dashed line) at ºF � 01. . The dotted line is the estimation
from [22] for the order parameter phase fluctuations.
by Yu.G. Pogorelov and the present authors in
Refs. 67–69. We, however, did not describe here this
important issue, since it deserves a special detailed re-
view.
5. Conclusions
In this paper, the BEC—BCS crossover from
superfluidity to superconductivity with doping in-
creasing at T � 0 in the cases of the s-wave and d-wave
pairing was briefly reviewed. In the 3D case this cross-
over does not take place at weak coupling constants,
and the same situation takes place in the d-wave pair-
ing case in two-dimensions, when the interaction does
not depend on the doping. When the correlation ra-
dius depends on the doping, the minimal value of cou-
pling for the two-particle bound state exists even in
the s-wave channel. Also the gap can decrease with the
doping in this case.
It was also shown, that the fluctuations of the or-
der parameter play an important role even at T � 0.
The fluctuations of the order parameter phase in weak
coupling limit in the case of the s-wave pairing regime
lead to increasing of superconductivity at any physical
carrier density, while the modulus fluctuations lead to
much stronger increasing of superconductivity at
small carrier densities. At large carrier densities they
lead to suppression of the order parameter, as the re-
sult, the gap decreases in BCS regime, when both the
modulus and the phase fluctuations are taken into ac-
count. This means that higher order fluctuation cor-
rections should be investigated in order to develop the
self-consistent theory of superconductivity (which can
be similar to superfluidity) at low carrier densities.
We would like to mention some direction which
can be interesting for future investigations. The study
of the problem of the crossover with realistic disper-
sion relations is not performed even on the mean-field
level in many interesting cases. The interplay between
disorder and superconductivity, as well strong correla-
tions and superconductivity in so-called bad metals
are another important problems. The fluctuations in
the d-wave pairing channel and in other nonisotropic
pairing channels even in the case of Gaussian fluctua-
tions were not studied carefully so far. It is also im-
portant to go beyond the Gaussian fluctuations, since
the pair susceptibility is diverging in the 2D and in
the 3D cases, as it was mentioned in Ref. 22. The role
of the interlayer coupling is another problem which is
not solved in general at the moment. The solution of
the problems mentioned above will lead to better un-
derstanding of the superconducting properties of sys-
tems with arbitrary carrier density and pairing poten-
tial.
Acknowledgments
First of all we would like to thank Prof. D.M. Ea-
gles who called our attention to the papers [3–5] and
kindly sent them to us. We also express our gratitude
to the referee of the survey whose constructive scien-
tific and editorial (mainly linguistic) remarks helped
us to improve it. One of us (V.M.L.) acknowledges
support by SCOPES-project 7UKPJ062150.00/1 of
the Swiss National Science Foundation.
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