Physica C 408–410 (2004) 355–357
www.elsevier.com/locate/physc
On intraband and interband BCS pairings in presence
of hybridization in two band superconductors
V.P. Ramunni
a,b
, G.M. Japiass
u c, A. Troper
b,d,*
a
d
Centro Atomico Constituyentes, Depto. Materiales, Av. General Paz 1499 (CP: 1650), San Martin,
Prov. de Buenos Aires, Argentina
b
Centro Brasileiro de Pesquisas Fısicas, Rua Dr. Xavier Sigaud, 150, Rio de Janeiro 22290-180, Brazil
c
Instituto de Fısica, Universidade Federal do Rio de Janeiro, Caixa Postal 68.528, Rio de Janeiro 21945-970, RJ, Brazil
Instituto de Fısica, Universidade do Estado do Rio de Janeiro, Rua S~ao Francisco Xavier, 524, Rio de Janeiro 20550-013, Brazil
Abstract
We discuss the stability of the intraband (r ¼ dd) and interband (r ¼ sd) couplings in the presence of a constant
hybridization on a two-band system. We calculate the critical temperature Tc;r associated to both type of couplings and
then we obtain the values of r ¼ 2jUr jDr ð0Þ=kB Tc;r and the isotope coefficients cr . One shows that under some circumstances, one can obtain values for r and cr which differ considerably from standard BCS couplings for both Cooper
pairings.
Ó 2004 Elsevier B.V. All rights reserved.
Keywords: Superconductivity; Hybridization; Isotope effect
1. Introduction
The effect of the electronic hybridization on narrow
band superconductivity has been extensively investigated in the literature [1]. In previous works [2,3], we
investigated the effect of one-body hybridization on the
superconductivity of a two-band model differently from
the standard approaches [1] for the uncoupled system
(we can only form Cooper pairing of one kind d–d or
s–d). Within the one-body formulation only single
electrons are transferred between different bands, so
that, for a local character hybridization, V ¼ V0 , we
verified that there is a critical value of the hybridization
Vc depending on a (which gives the ratio of the effective
band masses) and hD (Debye frequency) for which the
most of superconductor parameters vanishes. We obtain
the critical temperature Tc;r and the order parameter
D rð0Þ ¼ jUr jDr ð0Þ for the uncoupled system to calculate
the value of r ¼ D rð0Þ=kB Tc;r as a functions of V0 . We
also obtain an exact expression for the isotope coefficient cr directly from the BCS gap equation. In the absence of hybridization our results reproduce the
standard BCS results [4].
2. The Hamiltonian
The Hamiltonian describing our system is
H ¼ H0 þ Ha þ Hh ;
where
X ðsÞ y
X ðdÞ y
H0 ¼
Tij cir cjr þ
Tij dir djr ;
i;j;r
*
Corresponding author. Address: Centro Brasileiro de
Pesquisas Fısicas, Rua Dr. Xavier Sigaud, 150, Rio de Janeiro
22290-180, Brazil. Tel.: +55-21-2141-7285; fax: +55-21-21477400.
E-mail address: [email protected] (A. Troper).
ð1Þ
Ha ¼ Udd
X
ndir ndi;r þ Usd
X
i;r
Hh ¼
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.physc.2004.02.097
X
i;r
ð2Þ
i;j;r
Vii ðcyir dir þ diry cir Þ:
nsir ndi;r ;
ð3Þ
i;r
ð4Þ
V.P. Ramunni et al. / Physica C 408–410 (2004) 355–357
Dr
Ur
¼
8p
Z
hxD
qs ðzÞd
hxD
tanh
2
X
n¼1
b
xn ðzÞ
2
1.6
α = 0.1
α = 0.2
0
crða
mr ðzÞ Dr ð1Þn Cr V
Lnr ðzÞ ¼ 1 ð 1Þn
F ðzÞ
ð6Þ
1=2
1
;
½AðzÞ ð 1Þn F ðzÞ
2
ð7Þ
ð8Þ
2
2 2
2
2
F ðzÞ ¼ f½ða2 1Þz2 þ D2
dd þ 4V0 ½ða þ 1Þ z þ Ddd 2 2
2 1=2
þ 4D2
;
sd ½ða 1Þ z þ Ddd g
ð9Þ
where
8
r ¼ sd; r0 ¼ dd; Csd ¼ 2;
>
>
>
>
< msd ¼ ða 1Þ2 z2 þ 4V 2 þ D2 ;
dd
0
vmr ðzÞ ¼
0
>
¼
sd;
C
¼
4;
r
¼
dd;
r
>
dd
>
>
:
mdd ¼ ða2 1Þz2 þ 2V02 þ D2 ;
γsd / γBCS
0
-20
0
2
4
V0 / |Udd| . 10 -3
0.0
0.2
0.4
0.6
V0 / |Usd| . 10 -3
; V02 ; hD Þ
c P
nr ðzÞ
qs ð
hxD Þ 2n¼1 xL0;n
tanh bn2ðzÞ jz¼hxD
ðzÞ
¼
;
R hx
P
bc hxDD qs ðzÞ dz 2n¼1 Lnr ðzÞ 2 1 bcn ðzÞ
2
ð11Þ
xn ðzÞDr0 ;
2
AðzÞ ¼ ða2 þ 1Þz2 þ 2ðV02 þ D2
sd ÞDdd ;
1
D
cosh
and
xn ðzÞ ¼
D
α = 0.2 , θD
α = 0.2 , 2θ
20
Fig. 1. r ¼ 2Dr ð0Þ=kB Tc;r and isotope effect as functions of the
hybridization V0 =jUr j for several values of hD and the rate of a.
2
α = 0.1 , θD
α = 0.1 , 2θD
α = 0.2 , θD
α = 0.2 , 2θ
2
0
with
1
1.2
3
1
Lnr ðzÞ
xn ðzÞ
ð5Þ
α = 0.1
α = 0.2
2
ε*sd / εBCS
ε*dd / εBCS
H0 represents the (s) and (d) bands without interaction.
Ha represents a net attractive interaction between electrons of opposite spin at the same site (Ur < 0, r ¼ dd or
r ¼ sd) [5]. Hh is the mixing term which represents a oneparticle hybridization Hamiltonian in presence of the
periodic lattice. We take Vii ¼ V0 ¼ constant. The
Hamiltonian (4) is used to take into account the formation of both d–d and s–d pairings. We derive an exact
expression for the isotope coefficient directly from the
gap equation at T ¼ Tc , using the relation Tc M 1=2
within the BCS framework for any form of DOS. The
gap equation as a function of the absolute temperature
b ¼ 1=kB Tc is given by
γdd / γBCS
356
ð10Þ
2
being z ¼ E þ id and D2 ¼ 2D2
sd þ Ddd .
3. The isotope coefficient
For a local character hybridization Vii ¼ V0 , we derive
an exact expressions for the isotope coefficient ci 3 ¼
d lnTc;r =d lnM(r ¼ dd; sd) using the gap equation (5)
[2,4]. At the critical temperature ðUr Dr ðTc;r Þ ¼
Dr ðTc;r Þ ¼ 0Þ is given by 1 jUr jCr ¼ 0, using the relation xD M 1=2 , so that
where bcn ðzÞ ¼ bc x0n ðzÞ.
In Fig. 1, one gets for an arbitrary DOS, the following results: the value of r ¼ 2Dr ð0Þ=kB Tc;r differs
considerably from standard BCS result, in almost the
whole range of validity of the hybridization. We define
Vmin the value of a hybridization V0 where the superconducting parameters attain their minimum values.
Close to Vmin , dd increases as compared to the BCS result whereas Tc;dd tends towards zero. One sees that d–d
pairing favors an enhancement of dd , with increasing
hybridization, contrary to the case of s–d pairing, where
sd diminishes when V0 tends to a Vmin like Tc;sd does. For
both pairings the isotope coefficient cr exhibits the following features: close to Vmin , it reduces enormously as
compared with BCS result; whereas for very small V0 this
coefficient is the same as obtained in [4], and it takes a
maximum value for any value of V0 =jUr j depending on a
and hD (hD ¼ Debye frequency).
As a final comment: hybridization has a strong
connection with applied external pressure P in transition
metal like systems as well as in high Tc materials [6,7].
Under compression they can exhibit positive or negative
dTc =dP [8]. The analysis of this rather complicate
dependence is beyond the scope of this article and will be
discussed in a forthcoming work.
Acknowledgements
We would like to thank Conselho Nacional de Desenvolvimento Cientıfico e Tecnol
ogico––CNPq for
financial support. This work was partially performed
V.P. Ramunni et al. / Physica C 408–410 (2004) 355–357
under the frame
66201998-9.
of
PRONEX––project
number
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357
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