Physica C 384 (2003) 41–46
www.elsevier.com/locate/physc
Superconductivity in the periodic Anderson model
with anisotropic hybridization
L.G. Sarasua *, Mucio A. Continentino
Instituto de Fısica, Campus da Praia Vermelha, Universidade Federal Fluminense, Niter
oi, Rio de Janerio, 24.210-340, Brazil
Received 6 December 2001; received in revised form 1 May 2002; accepted 21 May 2002
Abstract
In this work we study superconductivity in the periodic Anderson model with both on-site and intersite hybridization, including the interband Coulomb repulsion. We show that the presence of the intersite hybridization together
with the on-site hybridization signiﬁcantly aﬀects the superconducting properties of the system. The symmetry of the
hybridization has a strong inﬂuence in the symmetry of the superconducting order parameter of the ground state. The
interband Coulomb repulsion may increase or decrease the superconducting critical temperature at small values of this
interaction, while is detrimental to superconductivity for strong values. We show that the present model can give rise to
positive or negative values of dTc =dP , depending on the values of the system parameters.
Ó 2002 Published by Elsevier Science B.V.
PACS: 74.62.Yb; 74.20.)z; 74.25.Dw
Keywords: Models for superconductivity; Superconducting phase diagrams; Pressure eﬀects
1. Introduction
Models in which correlated electrons are hybridized with a conduction band are of interest for
the understanding of the physical properties of
several systems [1–3]. Since the early proposal of
Anderson [4], the Hubbard and Anderson models
[5–7] were largely used as starting point for describing the normal and superconducting properties of high temperature superconductor cuprates
(HTSC). Despite the great amount of work that
has been done to determine the origin of the
*
Corresponding author. Tel.: +55-212-620-3881; fax: +55212-620-06735.
E-mail address: [email protected]ﬀ.br (L.G. Sarasua).
pairing mechanism in HTSC, the subject is still
focus of debate. For this reason, sometimes these
systems are modelled assuming a phenomenological attractive potential which includes all the
possible contributions for this coupling. The
emergence of superconductivity was largely studied in these models for diﬀerent symmetries of the
superconducting parameter [6,7]. Recently, it was
proposed that modiﬁcations in the hopping of the
Hubbard and extended Hubbard models are important to perform an accurate description and to
provide a mechanism for pairing [8,9]. These includes the consideration of more than nearestneighbor hoppings or correlations in the hoppings.
The eﬀect of hybridization on superconductivity
has been discussed recently [10,11]. However, a
common feature of the cited models is that the
0921-4534/02/$ - see front matter Ó 2002 Published by Elsevier Science B.V.
PII: S 0 9 2 1 - 4 5 3 4 ( 0 2 ) 0 1 8 3 9 - 7
42
L.G. Sarasua, M.A. Continentino / Physica C 384 (2003) 41–46
hopping and hybridization terms are assumed to
be isotropic. It is interesting to consider extensions
to the above models including diﬀerent symmetries
of the hybridization. In this work we study the
inﬂuence of the symmetry of the hybridization
on superconductivity for diﬀerent symmetries of
the superconducting order parameter, within the
framework of the periodic Anderson model with
an intersite attractive interaction, including both
on-site and intrasite hybridization. The eﬀect of
the Coulomb interaction between electrons of
diﬀerent bands on Tc is also considered. The work
is organized as follows. In Section 2 we present the
model. In Section 3 we obtain the critical temperature Tc and superconducting region for different symmetries of the hybridization for s- and
d-wave pairing. In Section 4 the eﬀect of the interband Coulomb repulsion on superconductivity
is considered. We also discuss the eﬀect of pressure
application on Tc in the present system. In Section
5 we summarize our results.
With the use of Hartree–Fock factorization in
(1), and performing a Fourier transformation, we
obtain the anomalous and standard GreenÕs
functions for the electrons of thef-band:
y
y
; fkr
iix ¼
hh fkr
y
iix ¼
hh fkr ; fkr
2
1
1
Dk ðx2 f0k ÞP ðxÞ
2p
ð2Þ
1
ðx f0k Þððx þ 0k Þðx þ f0k Þ Vk2 Þ
2p
P ðxÞ1
ð3Þ
with
P ðxÞ ¼ ððx 0k Þðx f0k Þ Vk2 Þ
2
ððx þ 0k Þðx þ f0k Þ Vk2 Þ D2k ðx2 f0k Þ;
where 0k ¼ k U hnfr i þ 0 , and k and f0k are the
energies of the unperturbed bands
k ¼ 2tf ðcosðkx aÞ þ cosðky aÞÞ;
f0k ¼ 2td ðcosðkx aÞ þ cosðky aÞÞ:
2. Hamiltonian model
We consider the periodic Anderson model in a
bidimensional square lattice
X y
X y
X
H ¼ tf
fir fjr þ td
dir djr U
nfi nfj
hijir
þ 0
X
hijir
nfi
þ
i
X
Vij ðfiry djr
hiji
þ
diry fjr Þ
ð1Þ
ijr
where firy ðfir Þ and diry ðdir Þ are the creation (annihilation) operators for f- and d-bands, nfi ¼ nfi" þ
nfi# , 0 is the site energy of f electrons, U is the
nearest-neighbor attraction between electrons of
the f-bands and the last term represents the hybridization between the bands. Here Vij includes
hybridization between electrons on the same site
and between electrons on nearest-neighbor sites.
We assume that Vij has an on-site-diagonal value
Vii ¼ V and oﬀ-diagonal values Vij ¼ Vx , Vy for
hybridization in the x- and y-directions. We introduce the superconducting order parameters
y y
fj;r i;
Dx ¼ U h fi;r
y y
Dy ¼ U h fi;r
fj0 ;r i
where j, j0 are neighbor sites of i, with ij and ij0
deﬁning directions parallel to x and y respectively.
so that the bandwidths of the f- and d-bands are
D ¼ 8tf and W ¼ 8td . We also introduced the notations
Vk ¼ V þ Vx cosðkx aÞ þ Vy cosðky aÞ;
Dk ¼ 2Dx cosðkx aÞ þ 2Dy cosðky aÞ:
In our study we shall concentrate in the cases of swave pairing (Dx ¼ Dy ) and d-wave pairing (Dx ¼
Dy ). In similar form, we specialize the hybridization to the cases Vx ¼ Vy and Vx ¼ Vy . We then
can express Dk and Vk as
Dk ¼ 2D cosðkx aÞ 2D cosðky aÞ
Vk ¼ V þ 2V 0 cosðkx aÞ 2V 0 cosðky aÞ V þ Vk0
ð4Þ
Thus, the hybridization can have a mixed symmetry, with one part being an (s-symmetric)
on-site hybridization, and the other an intersite
extended––s- or d-symmetric hybridization. In the
above equation we deﬁned Vk0 , as the k-dependent
part of the hybridization.
The roots of the polynomial P ðxÞ determine the
new quasiparticle energies of the system, which are
given by
L.G. Sarasua, M.A. Continentino / Physica C 384 (2003) 41–46
2
43
2
2
E1;2k
¼ 12ðD2k þ 2Vk2 þ 0k þ f0k Þ
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2 2
2
2
12 ðD2k þ 0k f0k Þ þ 4Vk2 ðD2k þ ð0k þ f0k Þ Þ
ð5Þ
In the following let us restrict to the half ﬁlled
band case (nf ¼ nd ¼ 1). We take the two bands 0k
and f0k to be centered at the Fermi level l ¼ 0
(symmetric case), which corresponds to set
0 ¼ U =2. We note that in the symmetric case the
presence of the on-site hybridization V opens a gap
in the middle of the band for any value of V [12] if
the f-band is localized. However, if the f-band has
a dispersion (tf 6¼ 0), as we suppose here, this gap
only arises if the on-site hybridization
pﬃﬃﬃﬃﬃﬃﬃﬃﬃ is larger
than a critical value Vg ¼ ðW =2Þ tf =td [13].
3. Inﬂuence of hybridization on superconductivity
From the propagators (2) and (3), we obtain the
following self-consistency equation which determine the value of D
"
2
2
1 X ck U D
ðE1k
f0k Þ
D¼
tanhðbE1k =2Þ
2
2
N k ðE1k
2E1k
E2k
Þ
#
2
2
ðE2k
f0k Þ
tanhðbE2k =2Þ
ð6Þ
2E2k
where ck ¼ cosðkx aÞ cosðky aÞ for s- and d-wave
pairing respectively and b ¼ 1=kB T . We have
solved Eq. (6) for various values of the model
parameters and diﬀerent symmetries of Dk and Vk0 .
Fig. 1 shows the superconducting region in the
plane (V and V 0 ) for d-wave pairing, with s- and dsymmetric Vk0 . The curves represent the points at
which the critical temperature Tc vanishes. Although superconductivity is always destroyed by
suﬃciently large values of V or V 0 , the superconducting region is very enlarged in certain directions
of the plane (V and V 0 ), for which resonance-like
peaks appear. The localization of these peaks depends on the hybridization and superconducting
parameter symmetries. In the case of d-symmetric
Vk0 and d-wave pairing, this resonance takes place
for directions deﬁned as V =V 0 ¼ constant 6¼ 0. In
contrast, for s-symmetric Vk0 the superconducting
Fig. 1. Superconducting phase diagram as a function of V and
V 0 for d-wave pairing. The curves represent the limit of the
superconducting region (SC) for U =W ¼ 0:25 (thin lines) and
U=W ¼ 0:5 (thick lines), with d-symmetric Vk0 (solid lines) and
s-symmetric Vk0 (dotted lines). In all the cases td =tf ¼ 2.
region extends almost along the axis V 0 . When the
pairing is s-wave the situation is very similar but
the positions of the peaks are altered. In fact, there
are two peaks for the s-wave pairing and s-symmetric Vk0 .
From Figs. 1 and 2 we can see that for given V
and V 0 , the two symmetries of the superconductor
parameter Dk may be favored, depending on the
symmetry of Vk0 and the values of the system parameters. However, there is a tendency to be the
most favorable situation those in which the hybridization and the superconductor parameter
have diﬀerent symmetries. For instance, for U ¼
0:25W , V ¼ 0:125W and V 0 ¼ 0:25W , the superconductor parameter is d-wave when Vk0 is s-symmetric, while is s-wave when Vk0 is d-symmetric.
When the hybridization has only the on-site component (V 0 ¼ 0), the critical temperature decays as
V is increased. In contrast, when V 0 is nonzero, Tc
may increase or decrease as V is increased. Fig. 3
shows the critical temperature Tc as a function of V
for two diﬀerent values of V 0 . From this, we see
that for vanishing V 0 , Tc is maximum at V ¼ 0 and
decreases monotonously with V. Instead, for
V 0 ¼ 0:06W , Tc is zero for small values of V. As V
is increased, Tc becomes diﬀerent from zero and
reaches a maximum for a ﬁnite value of V.
44
L.G. Sarasua, M.A. Continentino / Physica C 384 (2003) 41–46
in HTSC [14]. We conclude this section by noting
that, although the critical temperature can be increased by varying the ratio V =V 0 , the absolute
maximum value of Tc is obtained for V 0 ¼ V ¼ 0.
4. Interband repulsion and excitonic correlation
Let us consider now the eﬀect of the Coulomb
repulsion between f- and d-electrons on superconductivity. For this purpose we include the following term in the Hamiltonian (1)
X
Hint ¼
Gr;r0 ndir nfir0
ð7Þ
Fig. 2. Phase diagram for s-wave pairing, where the curves
correspond the same convention that in Fig. 1.
irr0
This is the so-called Falicov–Kimball term, which
has been extensively used to model valence transitions and metal–nonmetal transitions in mixed
valent compounds [15–17]. As in the previous
section we obtain the quasiparticle energies from
the poles of the GreenÕs functions
2
2
2
E1;2k
¼ 12ðD2k þ 2 Vek2 þ 0k þ f0k Þ
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
2
2
12 ðD2k þ 0k f0k Þ2 þ 4 Vek2 ðD2k þ ð0k þ f0k Þ2 Þ
ð8Þ
where we have deﬁned Vek ¼ Ve þ
Vk0 ,
with
Ve ¼ V þ AG
and
Fig. 3. The critical superconducting temperature Tc as a function of V for V 0 ¼ 0 (solid line) and V 0 ¼ 0:06W (dashed line),
(U ¼ 0:25W ; G ¼ 0, td =tf ¼ 2).
The results shown in Figs. 1 and 2 reveal the
strong inﬂuence of the hybridization symmetry on
the superconducting properties of the system. This
put in doubt results obtained from models in
which the isotropy of the overlap of the atomic
orbitals is assumed, and suggest that the symmetry
of the hybridization must be considered as part of
the problem, instead of ﬁxing it in an arbitrary
form. In actual systems, the anisotropy of the
hybridization can be originated in deviations of
the lattice constant or by a nonuniform distribution the p-orbitals orientation of the oxygen atoms
A ¼ hdiry fir i
is the excitonic parameter. Here 0k ¼ k U hnfr i þ
Ghndr i þ 0 , f0k ¼ fk þ Ghnfr i. In the above equations, we have assumed for simplicity that
Gr;r ¼ G, Gr;r ¼ 0. The hamiltonian H þ Hint is
equivalent to the spinless version of the Falicov–
Kimball model, for the case U ¼ 0 [16,17]. In the
mean ﬁeld approximation, the eﬀect of the interband Coulomb repulsion is to renormalize the
hybridization. For this reason, all the results of the
preceding section are still valid, but the value of V
must be replaced by its renormalized value Ve .
From the excitonic propagator
y
hhdkr ; fkr
iix ¼
1 e
1
V k ððx þ 0k Þðx þ f0k Þ Vek2 ÞP ðxÞ
2p
ð9Þ
L.G. Sarasua, M.A. Continentino / Physica C 384 (2003) 41–46
45
we obtain the self-consistency equation that determines the excitonic correlation A
"
2
Vek
1 X
ðE1k
þ 0k f0k Vek2 Þ
A¼
2
2
N k ðE1k
2E1k
E2k
Þ
tanhðbE1k =2Þ
ðE2 þ 0k f0k Vek2 Þ
2k
tanhðbE2k =2Þ
2E2k
#
ð10Þ
This must be solved together with (6), where now
the energies are given by (8). We have solved numerically the self-consistency equations (6) and
(10) for various values of G, V and V 0 . Fig. 4 shows
the V dependence of Ve . From this we can see that
Ve increases almost linear with V, with a slope
proportional to G. Grossly speaking, the eﬀect of
G is similar to multiplying V by a factor proportional to G. This causes the critical value Vc to
destroy superconductivity to reduce when G is
present (see Fig. 4). However, Tc can be enhanced
by G when there is present the intersite hybridization, as can be seen from Fig. 5. This occurs up
to moderate values of G. For higher values of G, Tc
is reduced to zero (Fig. 5). The f–d Coulomb interaction is always detrimental to superconductivity if there is present only the on-site component
Fig. 4. Eﬀective hybridization Ve as a function of V for
U ¼ 0:25W , td =tf ¼ 2 and (a) G ¼ 0:25W ; V 0 ¼ 0:2W , (b) G ¼
0:25W ; V 0 ¼ 0, (c) G ¼ 0:125; V 0 ¼ 0:2W and (d) G ¼ 0:125;
V 0 ¼ 0. The dashed lines indicate the values of V for which
superconductivity is destroyed.
Fig. 5. Superconducting critical temperature Tc as a function of
G=W for V ¼ 0:1W , U ¼ 0:25W , td =tf ¼ 2, with V 0 ¼ 0 (solid
line) and V 0 ¼ 0:06W (dotted line).
of the hybridization (V 0 ¼ 0), for any value of G.
We consider now the eﬀect of pressure application
P on the system. There are several systems for
which the critical temperature Tc is increased when
pressure is applied. It is thus of interest the development of models able to have negative or
positive values of dTc =dP in order to model these
systems. We show now that the present model can
have the two kind of behavior. The usual eﬀect of
pressure is the renormalization of the hopping integrals. In addition, the repulsive term G also can
be modiﬁed by the changes in lattice constants. We
expect that, in a general case, the values of the
derivatives of the parameters with respect to P;
dG=dP , dV =dP and dV 0 =dP do not have the same
values, due to their diﬀerent origins. Hence, the
values of Ve and V 0 will be renormalized in diﬀerent
manner by pressure application. In the preceding
section, it was shown that this may imply a increase or a reduction of the critical temperature Tc
depending on the values of the system parameters.
As a consequence, the present system is able to
have positive or negative values of dTc =dP . This is
caused by the approach or departure from the
optimal ratio value Ve =V 0 .
5. Conclusions
We have examined the superconducting properties of an extended periodic Anderson model with
46
L.G. Sarasua, M.A. Continentino / Physica C 384 (2003) 41–46
both intersite and intrasite hybridization, including
the Coulomb repulsion between f- and d-electrons.
The diagram of the superconducting region as a
function of the hybridization components V and
V 0 , for diﬀerent symmetries of the superconductor
order parameter and the intersite hybridization was
constructed. The obtained results show the importance of the hybridization symmetry and of the
presence of the components V and V 0 to determining the superconducting properties of the system and the symmetry of Dk in the ground state.
There are optimal values of the ratio V =V 0 for which
superconductivity is favored. We have considered
the eﬀect of the interband Coulomb repulsion G on
superconductivity. The result of our self-consistency calculation is that G is in general detrimental
to superconductivity, but may enhance the critical
temperature Tc up to moderate values of the interaction in the case in which there is present the intersite hybridization. The f –d Coulomb repulsion
is always detrimental to superconductivity for large
values of G or when there is only present the on-site
hybridization (V 0 ¼ 0). We have shown that the
present model can have positive or negative values
of dTc =dP , depending on the values of the model
parameters, which makes it useful to model compounds presenting the two signs of dTc =dP . We
conclude that the symmetry of the hybridization
and the presence of the intersite hybridization together with intrasite hybridization, are very important to determine relevant superconducting
properties and must been taken into account in the
description of superconducting systems.
Acknowledgements
We would like to thank Conselho Nacional de
Desenvolvimento Cientıﬁco e Tecnol
ogico-CNPq-
Brasil, Centro Latinoamericano de Fısica-CLAF,
and Facultad de Ciencias-Uruguay for partial ﬁnancial support.
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