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Hindawi Publishing CorporationAdvances in Condensed Matter PhysicsVolume 2013, Article ID 528960, 7 pageshttp://dx.doi.org/10.1155/2013/528960
Research ArticleThe Study on Hybridized Two-Band Superconductor
T. Chanpoom,1,2 J. Seechumsang,2,3 S. Chantrapakajee,4 and P. Udomsamuthirun2,3
1 Program of Physics and General Science, Faculty of Science and Technology, Rajabhat Nakhon Ratchasima University, Thailand2 Prasarnmitr Physics Research Unit, Department of Physics, Faculty of Science, Srinakharinwirot University, Sukhumvit 23,Bangkok 10110, Thailand
3Thailand Center of Excellence in Physics (ThEP), Si Ayutthaya Road, Bangkok 10400, Thailand4Rajamangala University of Technology Phra Nakhon, 399 Samsen Road, Dusit, Bangkok 10300, Thailand
Correspondence should be addressed to P. Udomsamuthirun; udomsamut55@yahoo.com
Received 3 December 2012; Revised 8 March 2013; Accepted 22 March 2013
Academic Editor: Victor V. Moshchalkov
Copyright ยฉ 2013 T. Chanpoom et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The two-band hybridized superconductor which the pairing occurred by conduction electron band and other-electron band areconsidered within a mean-field approximation. The critical temperature, zero-temperature order parameter, gap-to-๐
๐ratio, and
isotope effect coefficient are derived. We find that the hybridization coefficient shows a little effect on the superconductor thatconduction electron band has the same energy as other-electron band but showsmore effect on the superconductor that conductionelectron band coexists with lower-energy other-electron band.The critical temperature is decreased as the hybridization coefficientincreases. The higher value of hybridization coefficient, lower value of gap-to-๐
๐ratio, and higher value of isotope effect coefficient
are found.
1. Introduction
Since Moskalenko [1, 2] Suhl et al. [3] introduced the two-band model that accounts for multiple energy bands in thevicinity of the Fermi energy contributing electron pairingin superconductor, the two-band model has been applied tohigh temperature superconductor in copper oxides [4โ10],MgB2superconductor [11โ13], and heavy Fermion supercon-
ductor [14, 15]. Dolgov et al. [16] studied the thermodynamicproperties of the two-band superconductor: MgB
2. The
superconducting energy gap, free energy, the entropy, andheat capacity were calculated within the framework of two-band Eliashberg theory. Mazin et al. [17] studied the effect ofinterband impurity scattering on the critical temperature oftwo-band superconductor in MgB
2. Askerzade and Tanatar
[18] and Changjan and Udomsamuthirun [19] calculated thecritical field of the two-band superconductor by Ginzburg-Landau approach and applied it to Fe-based superconduc-tors. Golubov and Koshelev [20] investigated the two-bandsuperconductor with strong intraband and weak interbandelectronic scattering rates in the framework of coupledUsadelequation.
The interplay of superconductivity and magnetism is oneof the most interesting phenomena of superconductor. Thecuprate superconductor exhibits the phase diagram havingthe magnetic ordered states in the vicinity of the super-conducting phase. The antiferromagnetic and ferromagneticphases are also found in the heavy Fermion superconductor.In ErRh
4B6[21] and HoMo
6S8[22] system, ๐ -wave super-
conductivity shows ferromagnetism in the ground state atintermediate temperature.
The nesting properties of Fermi surface in low dimen-sional system arise the spin density wave (SDW) state andcharge density wave (CDW) state in the interplay of super-conductivity andmagnetism system.The SDWand the CDWstates occurred by Coulomb interaction between electron,and electron-phonon interaction, respectively. Nass et al. [23]used the BCS-type pairing to explain the antiferromagneticsuperconductor. Suzumura and Nagi [24] investigated someproperties of antiferromagnetic superconductor. They pro-posed the Hamiltonian of the superconductivity associatedwith conduction ๐-electron and the antiferromagnetismassociatedwith๐-electron of rare earth atoms that formed theBCS-type pairing. Ichimura et al. [25] investigated the effect
2 Advances in Condensed Matter Physics
of the CDW on BCS superconductor within a mean-fieldapproximation. In this model, the two order parametersof CDW and superconductor were introduced. The tight-binding band in 2D square lattice and nesting vector ๐ =
(๐, ๐) was used in their calculations. Rout and Das [26]applied the periodic Anderson model (PAM) for calculatingthe nonmagnetic ground state of heavy Fermion super-conductor. The Hamiltonian of the heavy Fermion systemscomposed of conduction electron band, ๐-electron band,the hybridization of conduction electron and ๐-electronband, BCS-like pairing band, and the intra-atomic Coulombinteraction of ๐-electron. The Hamiltonian was simplifiedby linearising the intra-atomic Coulomb interaction withthe Hartree-Fock approximation then the ๐-electron bandenergy was ๐ธ
๐= ๐๐
+ ๐๐๐, where ๐
๐was bare ๐-electron
energy and ๐๐๐was the Coulomb energy of ๐-electron.
Finally, their Hamiltonian were consisted of conductionand ๐-electron band including BCS-like pairing band andhybridization term. Panda andRout [27] studied the interplayof CDW, SDW, and superconductivity in high temperaturesuperconductor in low doping phase. The model of mean-field Hamiltonian including the CDW, SDW and supercon-ductivity was introduced.
In this paper, we modified the hybridized Hamiltonianof Rout and Das [26] to be the two-band superconductorwith hybridization. Some physical properties of two-bandhybridized superconductor, that is, critical temperature, zero-temperature order parameter, gap-to-๐
๐ratio, and isotope
effect coefficient, were investigated.
2. Model and Calculation
According to the hybridized Hamiltonian [26] that consistedof conduction electron and๐-electron band, BCS-like pairingband, and hybridization term, we have
๐ป = โ
๐,๐
๐๐๐ถ+
๐๐๐ถ๐๐
+ ๐๐
โ
๐,๐
๐+
๐๐๐๐๐
+ ๐พ0
โ
๐,๐
(๐+
๐๐๐ถ๐๐
+ ๐ถ+
๐๐๐๐๐
)
+๐
2โ
๐,๐
๐๐
๐๐๐๐
๐,โ๐โ ฮ โ
๐
(๐ถ+
๐โ๐ถ+
โ๐โ+ ๐ถโ๐โ
๐ถ๐โ
) ,
(1)
where ๐ถ+
๐๐(๐ถ๐๐
) and ๐+
๐๐(๐๐๐
) are the creation (annihilation)operator of conduction electron and ๐-electron. ฮ is thesuperconducting order parameter that Cooper pairs involveonly the conduction electron. ๐พ
0is the hybridization inter-
action coefficient of ๐-electron band and conduction band.๐๐
๐= ๐+
๐๐๐๐๐is the intraatomic Coulomb interaction between
๐-electron. They [26] linearised the (๐/2) โ๐,๐
๐๐
๐๐๐๐
๐,โ๐term
by Hartree-Fock approximation that (๐/2) โ๐,๐
๐๐
๐๐๐๐
๐,โ๐โ
๐ โ๐,๐
๐โ๐
๐+
๐๐๐๐๐; then the Hamiltonian became
๐ป = โ
๐,๐
๐๐๐ถ+
๐๐๐ถ๐๐
+ โ
๐,๐
๐ธ0๐+
๐๐๐๐๐
+ ๐พ0
โ
๐,๐
(๐+
๐๐๐ถ๐๐
+ ๐ถ+
๐๐๐๐๐
)
โ ฮ โ
๐
(๐ถ+
๐โ๐ถ+
โ๐โ+ ๐ถโ๐โ
๐ถ๐โ
) ,
(2)
where ๐ธ0
= ๐๐
+ ๐๐โ๐
that is the energy collected the non-interaction with a modified ๐-level.
In our model, the two-band superconductor comprise ofconduction electron and other-electron band. The supercon-ducting order parameters can occurr by conduction electronand other-electron band. The conduction band makes theintra-atomic Coulomb interaction with other-electron band.We set
๐ป = โ
๐,๐
๐๐๐ถ+
๐๐๐ถ๐๐
+ โ
๐,๐
๐ธ0๐+
๐๐๐๐๐
+ ๐พ0
โ
๐,๐
(๐+
๐๐๐ถ๐๐
+ ๐ถ+
๐๐๐๐๐
)
+๐
2โ
๐,๐
๐๐
๐๐๐๐
๐,โ๐โ ฮ โ
๐
(๐ถ+
๐โ๐ถ+
โ๐โ+ ๐ถโ๐โ
๐ถ๐โ
)
โ ฮ โ
๐
(๐+
๐โ๐+
โ๐โ+ ๐โ๐โ
๐๐โ
) ,
(3)
where ๐ถ+
๐๐(๐ถ๐๐
) and ๐+
๐๐(๐๐๐
) are the creation (annihilation)operators of conduction electron and other-electron band. ฮis the superconducting order parameter. ๐พ
0is the hybridiza-
tion interaction coefficient of other-electron band and con-duction band. ๐๐
๐= ๐+
๐๐๐๐๐is the intra-atomic Coulomb inter-
action between other-electron.The Hamiltonian is linearisedby [26]โs technique; then we get the simplified two-band-hybridized Hamiltonian. We can write the Hamiltonian as
๐ป = ๐ป1
+ ๐ป2
+ ๐ป12
, (4)
where
๐ป1
= โ
๐๐
๐๐๐ถ+
๐๐๐ถ๐๐
โ ฮ โ
๐
(๐ถ+
๐โ๐ถ+
โ๐โ+ ๐ถโ๐โ
๐ถ๐โ
) , (5a)
๐ป2
= โ
๐๐
๐ธ๐๐+
๐๐๐๐๐
โ ฮ โ
๐
(๐+
๐โ๐+
โ๐โ+ ๐โ๐โ
๐๐โ
) , (5b)
๐ป12
= ๐พ0
โ
๐๐
(๐+
๐๐๐ถ๐๐
+ ๐ถ+
๐๐๐๐๐
) . (5c)
The first Hamiltonian describes the conduction electronHamiltonian, the second Hamiltonian describes the Hamil-tonian of other-electron band, and the third Hamiltoniandescribes the interact Hamiltonian. Where ๐
๐and ๐ธ
๐are the
band energies of the conduction electron and other-electronband measured from the Fermi energy.
๐ธ๐
= ๐ธ0+๐๐โ๐
is the energy collected the non-interactionwith a modified other-electron band. ๐ถ
+
๐๐(๐ถ๐๐
) and ๐+
๐๐(๐๐๐
)
are the creation (annihilation) operator of conduction elec-tron and other-electron. ๐พ
0is the hybridization interaction
coefficient of other-electron band and conduction band. ฮ isthe effective superconducting order parameter occurred byconduction electron and other-electron and assumed to behomogeneous in space. The effective superconducting orderparameters is
ฮ =๐
2โ
๐
(โจ๐ถ+
๐โ๐ถ+
โ๐โโฉ + โจ๐
+
๐โ๐+
โ๐โโฉ) , (6)
where the ๐ -wave like BCS pairing interaction having thesame coupling interaction potential is assumed. The effective
Advances in Condensed Matter Physics 3
superconducting order parameters are the coupling equationof conduction electron and other-electron that can occurr inmagnetic superconductor [23โ25]. However for simplicity ofcalculation, the same pairing strength is taken [28].
We introduce the finite-temperature Green function:
๐บ (๐, ๐) = โ โจ๐๐๐๐
(๐) ๐+
๐(0)โฉ , (7)
where ๐+
๐= (๐ถ
+
๐โ, ๐ถโ๐โ
, ๐+
๐โ, ๐โ๐โ
) and ๐๐is the ordering
operator for imaginary time, ๐ = ๐๐ก.After some calculations, the Green function in Nambu
representation is obtained:
๐บ (๐๐, ๐) = (๐๐
๐โ (
๐๐
โ ๐ธ๐
2) ๐3๐3
โ (๐๐
+ ๐ธ๐
2) ๐3
+ ฮ๐1
โ ๐พ0๐1๐3)
โ1
,
(8)
where๐๐
= ๐๐(2๐+1),๐ is temperature, ๐ is an integer, and๐๐
and ๐๐
(๐ = 1, 2, 3) are the Pauli matrices. Our Green functionobtained shows the same form as [25] that investigated theeffect of the CDW on BCS superconductor within a mean-field approximation, ๐บ
โ1
(๐๐, ๐) = ๐๐
๐โ ๐พ๐๐3๐3
โ ๐ฟ๐๐0๐3
+
ฮ๐0๐1
+ ๐ค๐1๐3. All parameters detailed can be found in [25].
We find that (๐๐
โ ๐ธ๐)/2 โก ๐พ
๐and (๐
๐+ ๐ธ๐)/2 โก ๐ฟ
๐, where ๐พ
๐
and ๐ฟ๐are the band structure energies in 2D square lattice of
nearest-neighbor and next-nearest-neighbor transfer, respec-tively. And โ๐พ
0โก ๐ค which ๐ค is the order parameter of CDW.
This result means that the CDWconsideration gives the sameresult as the hybridization consideration within a mean-fieldapproximation.
From (6) and (8), the superconducting gap equation is
1
๐=
1
4โ
๐
(
tanh (โฮ2 + ๐2โ/2๐)
โฮ2 + ๐2โ
+
tanh (โฮ2 + ๐2+/2๐)
โฮ2 + ๐2+
) ,
(9)
where ๐+
= (๐๐
+ ๐ธ๐)/2 + โ((๐
๐โ ๐ธ๐)/2)2
+ ๐พ2
0and ๐
โ=
(๐๐+๐ธ๐)/2โโ((๐
๐โ ๐ธ๐)/2)2
+ ๐พ2
0.The ๐
+and ๐โrepresent the
upper and lower bands of quasiparticle energy spectra of thehybridization system.We can determine the superconductingcritical temperature ๐
๐by putting ฮ โ 0; then
1
๐=
1
4โ
๐
(tanh (๐
โ/2๐๐)
๐โ
+tanh (๐
+/2๐๐)
๐+
) . (10)
In the absence of the hybridization interaction, ๐พ0
= 0; thatis,
1
๐=
1
2โ
๐
(tanh (๐/2๐
๐0)
๐) or 1
๐= ln(
2๐พ๐๐ท
๐๐๐0
) ,
(11)
where ๐พ = 1.78. ๐ = ๐(0)๐, ๐ is the coupling constant, and๐(0) is the constant density of state at the Fermi surface, and
๐๐ทis theDebye cutoff energy.๐
๐0is the critical temperature of
superconductor without hybridization that the BCSโs result.Because of the complicated quasi-particle energy spectra
obtained, we introduce the two approximated conditions tocalculate analytically; the superconductor with conductionelectron band having the same energy as other-electron band(๐๐
โ ๐ธ๐) and the superconductor with conduction electron
band coexistingwith lower-energy other-electron band (๐๐
โซ
๐ธ๐, ๐ธ๐
โ 0).
Case 1. The superconductor that the conduction electronband having the same energy as other-electron band.
In this case, the approximation is ๐๐
โ ๐ธ๐. Then, we get
๐โ
โ ๐๐
โ ๐พ0and ๐+
โ ๐๐
+ ๐พ0. The difference of the lower
and upper energy spectra is equal to 2๐พ0. If ๐พ0
= 0, theBCSโs superconductor is obtained. In this case, the effect ofthe hybridization interaction on the BCS superconductor isconsidered.
We substitute above approximations into (10); then thegap equation becomes
1
๐=
1
4(โซ
๐๐ท
โ๐๐ท
tanh (๐โ/2๐๐)
๐โ
๐๐๐
+ โซ
๐๐ท
โ๐๐ท
tanh (๐+/2๐๐)
๐+
๐๐๐)
โ ln(
2๐พโ๐2
๐ทโ ๐พ2
0
๐๐๐
) .
(12)
The critical temperature is
๐๐
= 1.13โ๐2
๐ทโ ๐พ2
0๐โ1/๐
. (13)
And the zero-temperature energy gap can be found as
1
๐=
1
4โซ
๐๐ท
โ๐๐ท
(1
โฮ2 (0) + (๐๐
โ ๐พ0)2
+1
โฮ2 (0) + (๐๐
+ ๐พ0)2
) ๐๐๐
=1
2(sin hโ1 (
๐๐ท
โ ๐พ0
ฮ (0)) + sin hโ1 (
๐๐ท
+ ๐พ0
ฮ (0))) .
(14)
For ๐๐ท
โซ ฮ(0), we can get
ฮ (0) = 2โ๐2
๐ทโ ๐พ2
0๐โ1/๐
. (15)
From (13) and (15), the gap-to-๐๐ratio is obtained:
๐ =2ฮ (0)
๐๐
= 3.53. (16)
In this case, we find that the hybridization interaction coeffi-cient decreases the critical temperature and zero-temperatureenergy gap but has no effect on gap-to-๐
๐ratio.
4 Advances in Condensed Matter Physics
To investigate the effect of hybridization and the Debyecutoff on gap-to-๐
๐ratio, we rewrite the gap equation at crit-
ical temperature and at zero-temperature into the form [29]
โซ
(๐๐ท+๐พ0)/(2๐๐)
(โ๐๐ท+๐พ0)/(2๐๐)
tanh๐ฅ
๐ฅ๐๐ฅ = โซ
2(๐๐ท+๐พ0)/๐๐
โ2(๐๐ทโ๐พ0)/๐๐
1
โ๐ 2 + ๐ฅ2๐๐ฅ.
(17)
The numerical calculation of (17) is shown in Figure 2.Within the definition of isotope effect coefficient in
harmonic approximation; ๐ผ = (1/2)(๐๐ท
/๐๐)(๐๐๐/๐๐๐ท
) andequation (9), we
๐ผ =๐๐ท
2(tanh (๐
โ
๐ท/2๐๐) /๐โ
๐ท+ tanh (๐
+
๐ท/2๐๐) /๐+
๐ท
tanh (๐โ
๐ท/2๐๐) + tanh (๐
+
๐ท/2๐๐)
) ,
(18)
where ๐+
๐ท= ๐๐ท
+ ๐พ0and ๐
โ
๐ท= ๐๐ท
โ ๐พ0.
Consider the limiting cases that the hybridization is sosmall with respect to Debye cutoff energy, ๐
๐ทโซ ๐พ0; for ๐
๐ท>
2๐๐, we can get that ๐ผ โ (๐
๐ท/4)(1/๐
โ
๐ท+ 1/๐
+
๐ท) โ 1/2, and for
๐๐ท
< 2๐๐, we get ๐ผ โ (๐
๐ท/2)((1/๐
๐)/(๐๐ท
/๐๐)) โ 1/2 that the
BCSโ result.
Case 2. The superconductor that the conduction electronband coexisting with lower-energy other-electron band.
Because of the hybridization Hamiltonian having thesame Greenโs function as the charge density wave model, wecan apply this model to the superconducting state found inheavy Fermion superconductor. The heavy Fermion super-conductor has its origin in the interplay of strong Coulombrepulsion in 4f- and 5f-shells and their hybridizations withthe conduction band. The ๐-electron is associated with themagnetic ordering having lower energy than conductionelectron. We can make the assumption that the ๐-electronband is at the Fermi level which can be taken as ๐ธ
๐โ 0; then
๐โ
โ๐๐
2โ ๐พ0, ๐
+โ
๐๐
2+ ๐พ0, for
๐๐
2< ๐พ0,
๐โ
โ 0, ๐+
โ ๐๐, for
๐๐
2> ๐พ0.
(19)
Substituting above approximation into (10), we can get
1
๐=
1
4(โซ
๐๐ท
โ๐๐ท
tanh (๐โ/2๐๐)
๐โ
๐๐๐
+ โซ
๐๐ท
โ๐๐ท
tanh (๐+/2๐๐)
๐+
๐๐๐)
โ ln(2๐พ
๐๐๐
โ2๐๐ท
๐พ0) +
1
2(
๐๐ท
2๐๐
โ๐พ0
๐๐
) ,
(20)
where,
โซ
๐๐ท
โ๐๐ท
tanh (๐โ/2๐๐)
๐โ
๐๐๐
โ 2 (๐๐ท
2๐๐
โ2๐พ0
2๐๐
) + 2 ln(4๐พ๐พ0
๐๐๐
) ,
โซ
๐๐ท
โ๐๐ท
tanh (๐+/2๐๐)
๐+
๐๐๐
โ 2 ln(2๐พ๐๐ท
๐๐๐
) .
(21)
The critical temperature is
๐๐
= 1.13โ2๐พ0๐๐ท
๐โ1/๐+(1/2)(๐๐ท/2๐๐โ๐พ0/๐๐). (22)
The gap equation as zero-temperature is
1
๐=
1
4โซ
๐๐ท
โ๐๐ท
(1
โฮ2 (0) + (๐โ)2
+1
โฮ2 (0) + (๐+)2
) ๐๐๐,
(23)
where,
โซ
๐๐ท
โ๐๐ท
1
โฮ2 (0) + (๐โ)2
๐๐๐
โ (2
ฮ (0)) (๐๐ท
โ 2๐พ0) + 2 sin hโ1 (
2๐พ0
ฮ (0)) ,
โซ
๐๐ท
โ๐๐ท
1
โฮ2 (0) + (๐+)2
๐๐๐
โ 2 sin hโ1 ( ๐๐ท
ฮ (0)) .
(24)
Then, we get
ฮ (0) = 2โ2๐พ0๐๐ท
๐โ1/๐+(1/2)((๐๐ทโ2๐พ0)/ฮ(0)). (25)
According to (22) and (25), the gap-to-๐๐ratio is
๐ =2ฮ (0)
๐๐
= 3.53๐((๐๐ทโ2๐พ0)/๐๐)(1/๐ โ1/4). (26)
To investigate the effect of hybridization and the Debyecutoff on gap-to-๐
๐ratio, we rewrite the gap equation at crit-
ical temperature and at zero-temperature into the form [29]
โซ
๐๐ท/๐๐
โ๐๐ท/๐๐
tanh(๐ฆ/4 โ โ(๐ฆ/4))
๐ฅ๐๐ฅ
= โซ
2(๐๐ท+๐พ0)/๐๐
โ2(๐๐ทโ๐พ0)/๐๐
1
โ๐ 2 + ๐ฅ2๐๐ฅ.
(27)
The numerical calculation of this equation is shown inFigure 2.
Within the definition of isotope effect coefficient in har-monic approximation, (10) and ๐ธ
๐โ 0, we can get
๐ผ = (๐๐ท
2)
ร (1 + (2๐
๐/๐๐ท
) tanh (๐๐ท
/2๐๐)
๐๐ท
โ 2๐พ0
+ 2๐๐(tanh (๐
๐ท/2๐๐) + tanh (๐พ
0/๐๐))
) .
(28)
3. Results and Discussions
We use the hybridized two-band Hamiltonian to investigatethe critical temperature, zero-temperature order parameter,gap-to-๐
๐ratio, and isotope effect coefficient of superconduc-
tor. The Green function and gap equation are derived ana-lytically. However, the quasi-particle energy spectra obtained
Advances in Condensed Matter Physics 5๐๐
40
30
20
10
00 1 2 3 4
๐พ0/๐๐
Case 1๐๐ท = 300 ๐ = 0.3Case 1๐๐ท = 300 ๐ = 0.2Case 1๐๐ท = 200 ๐ = 0.2
Case 2 ๐๐ท = 200 ๐ = 0.2Case 2 ๐๐ท = 300 ๐ = 0.2Case 2 ๐๐ท = 300 ๐ = 0.3
Figure 1: The ๐๐versus hybridization coefficients of Cases 1 and 2.
are complicated; then we introduce two approximated casesas, the superconductor that conduction electron band hasthe same energy as other-electron band (๐
๐โ ๐ธ๐) and the
superconductor that conduction electron band coexists withlower-energy other-electron band (๐
๐โซ ๐ธ๐, ๐ธ๐
โ 0). Thecritical temperature, zero-temperature order parameter, gap-to-๐๐ratio, and isotope effect coefficient are shown in the
exact forms.We use the integration forms of involved equations for
more accuracy in numerical calculation. After the numericalcalculations of ๐
๐, (Figure 1), we find that the weak-coupling
limit (๐ < 0.4) can be found in range of ๐๐ท
/๐๐
> 10. Then,we investigate the effect of hybridization in weak-couplinglimit with ๐
๐ท/๐๐
> 10 and ๐พ0/๐๐
= 0.2, 4.0. We find that ๐๐
is decreased when the hybridization coefficient increased inCase 2 and no effect in Case 1. In Figure 2, the gap-to-๐
๐ratios
(๐ ) of Cases 1 and 2 with varied hybridization coefficientsare shown. In Case 1, the ๐ is tended to BCS value, 3.53, as๐๐ท
/๐๐
โ โ. Case 2, ๐ โ 3.3โ3.9 can be found to dependstrongly on the value of ๐พ
0/๐๐. The higher value of ๐พ
0/๐๐and
the lower value of ๐ are found to agree with superconductinggapโs behavior of HF system of Rout et al. [30].
The isotope effect coefficient is also investigated andshown in Figure 3. In Case 2, we find that the isotope effectcoefficient can be more than the BCS (๐ผ > 0.5) and less thanBCS value (๐ผ < 0.5), but inCase 1we can find only for๐ผ > 0.5.However, ๐ผ in both cases is converse to 0.5 as ๐
๐ท/๐๐
โ โ.The Fe-based superconductors are mutiband system
which comprises at least two bands which propose ๐ -waveparing state. The ๐ = 3.68 [31] which shows a consistentmanner with the BCS prediction is found. The value of iso-tope effect coefficient were found to be ๐ผ โ 0.35โ0.4 [32] and๐ผFe = 0.81 [33]. These results indicate that electron-phononinteraction plays some role in the superconducting mecha-nism by affecting the magnetic properties [31]. According toour model, the effective superconducting order parameters
4.0
3.8
3.6
3.4
3.2
3.0
๐
10 12 14 16 18 20๐๐ท/๐๐
Case 2 ๐พ0/๐๐ = 4.0Case 1๐พ0/๐๐ = 4.0
Case 2 ๐พ0/๐๐ = 0.2Case 1๐พ0/๐๐ = 0.2
Figure 2: The gap-to-๐๐ratio of Cases 1 and 2 with varied
hybridization coefficients.
๐ผ
1.00
0.75
0.50
๐๐ท/๐๐
10 12 14 16 18 20
Case 2 ๐พ0/๐๐ = 4.0Case 1๐พ0/๐๐ = 4.0
Case 2 ๐พ0/๐๐ = 0.2Case 1๐พ0/๐๐ = 0.2
Figure 3: The isotope effect coefficient of Cases 1 and 2 with variedhybridization coefficients.
are the coupling equation of conduction electron and other-electron and the magnetic order are included in our calcu-lation as in Case 2. We can get the experimental data of Fe-based superconductor from Figures 2 and 3.
4. Conclusions
The two-band hybridized superconductor that the pairingoccurred by conduction electron band and other-electronband is studied inweak-coupling limit.The formula of criticaltemperature, zero-temperature order parameter, gap-to-๐
๐
ratio and isotope effect coefficient are calculated. For thesuperconductor that conduction electron band has the sameenergy as other-electron band, the hybridization coefficient
6 Advances in Condensed Matter Physics
shows a little effect. The numerical results do not differ muchfrom the BCSโs results. For the superconductor that conduc-tion electron band coexists with lower-energy other-electronband, the hybridization coefficient show more effect. ๐
๐is
decreased when the hybridization coefficient increases. Wecan get higher and lower value of ๐ and ๐ผ than BCSโs resultsdepending on the hybridization coefficient. Higher value ofhybridization coefficient, lower value of ๐ , and higher valueof ๐ผ are found.
Acknowledgments
The financial support of the Office of the Higher EducationCommission, Srinakhariwirout University, and ThEP Centeris acknowledged.
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