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; [email protected]
Received 3 December 2012; Revised 8 March 2013; Accepted 22 March 2013
Academic Editor: Victor V. Moshchalkov
Copyright Β© 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-π
πratio, 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-π
πratio, 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 [4β10],MgB2superconductor [11β13], 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, π -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 π-electron and the antiferromagnetismassociatedwithπ-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 π =
(π, π) 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, π-electron band,the hybridization of conduction electron and π-electronband, BCS-like pairing band, and the intra-atomic Coulombinteraction of π-electron. The Hamiltonian was simplifiedby linearising the intra-atomic Coulomb interaction withthe Hartree-Fock approximation then the π-electron bandenergy was πΈ
π= ππ
+ πππ, where π
πwas bare π-electron
energy and πππwas the Coulomb energy of π-electron.
Finally, their Hamiltonian were consisted of conductionand π-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-π
πratio, and isotope
effect coefficient, were investigated.
2. Model and Calculation
According to the hybridized Hamiltonian [26] that consistedof conduction electron andπ-electron band, BCS-like pairingband, and hybridization term, we have
π» = β
π,π
πππΆ+
πππΆππ
+ ππ
β
π,π
π+
πππππ
+ πΎ0
β
π,π
(π+
πππΆππ
+ πΆ+
πππππ
)
+π
2β
π,π
ππ
ππππ
π,βπβ Ξ β
π
(πΆ+
πβπΆ+
βπβ+ πΆβπβ
πΆπβ
) ,
(1)
where πΆ+
ππ(πΆππ
) and π+
ππ(πππ
) are the creation (annihilation)operator of conduction electron and π-electron. Ξ is thesuperconducting order parameter that Cooper pairs involveonly the conduction electron. πΎ
0is the hybridization inter-
action coefficient of π-electron band and conduction band.ππ
π= π+
πππππis the intraatomic Coulomb interaction between
π-electron. They [26] linearised the (π/2) βπ,π
ππ
ππππ
π,βπterm
by Hartree-Fock approximation that (π/2) βπ,π
ππ
ππππ
π,βπβ
π βπ,π
πβπ
π+
πππππ; then the Hamiltonian became
π» = β
π,π
πππΆ+
πππΆππ
+ β
π,π
πΈ0π+
πππππ
+ πΎ0
β
π,π
(π+
πππΆππ
+ πΆ+
πππππ
)
β Ξ β
π
(πΆ+
πβπΆ+
βπβ+ πΆβπβ
πΆπβ
) ,
(2)
where πΈ0
= ππ
+ ππβπ
that is the energy collected the non-interaction with a modified π-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
π» = β
π,π
πππΆ+
πππΆππ
+ β
π,π
πΈ0π+
πππππ
+ πΎ0
β
π,π
(π+
πππΆππ
+ πΆ+
πππππ
)
+π
2β
π,π
ππ
ππππ
π,βπβ Ξ β
π
(πΆ+
πβπΆ+
βπβ+ πΆβπβ
πΆπβ
)
β Ξ β
π
(π+
πβπ+
βπβ+ πβπβ
ππβ
) ,
(3)
where πΆ+
ππ(πΆππ
) and π+
ππ(πππ
) are the creation (annihilation)operators of conduction electron and other-electron band. Ξis the superconducting order parameter. πΎ
0is the hybridiza-
tion interaction coefficient of other-electron band and con-duction band. ππ
π= π+
πππππis the intra-atomic Coulomb inter-
action between other-electron.The Hamiltonian is linearisedby [26]βs technique; then we get the simplified two-band-hybridized Hamiltonian. We can write the Hamiltonian as
π» = π»1
+ π»2
+ π»12
, (4)
where
π»1
= β
ππ
πππΆ+
πππΆππ
β Ξ β
π
(πΆ+
πβπΆ+
βπβ+ πΆβπβ
πΆπβ
) , (5a)
π»2
= β
ππ
πΈππ+
πππππ
β Ξ β
π
(π+
πβπ+
βπβ+ πβπβ
ππβ
) , (5b)
π»12
= πΎ0
β
ππ
(π+
πππΆππ
+ πΆ+
πππππ
) . (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 π
πand πΈ
πare the
band energies of the conduction electron and other-electronband measured from the Fermi energy.
πΈπ
= πΈ0+ππβπ
is the energy collected the non-interactionwith a modified other-electron band. πΆ
+
ππ(πΆππ
) and π+
ππ(πππ
)
are the creation (annihilation) operator of conduction elec-tron and other-electron. πΎ
0is the hybridization interaction
coefficient of other-electron band and conduction band. Ξ isthe effective superconducting order parameter occurred byconduction electron and other-electron and assumed to behomogeneous in space. The effective superconducting orderparameters is
Ξ =π
2β
π
(β¨πΆ+
πβπΆ+
βπββ© + β¨π
+
πβπ+
βπββ©) , (6)
where the π -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 [23β25]. However for simplicity ofcalculation, the same pairing strength is taken [28].
We introduce the finite-temperature Green function:
πΊ (π, π) = β β¨ππππ
(π) π+
π(0)β© , (7)
where π+
π= (πΆ
+
πβ, πΆβπβ
, π+
πβ, πβπβ
) and ππis the ordering
operator for imaginary time, π = ππ‘.After some calculations, the Green function in Nambu
representation is obtained:
πΊ (ππ, π) = (ππ
πβ (
ππ
β πΈπ
2) π3π3
β (ππ
+ πΈπ
2) π3
+ Ξπ1
β πΎ0π1π3)
β1
,
(8)
whereππ
= ππ(2π+1),π is temperature, π is an integer, andππ
and ππ
(π = 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, πΊ
β1
(ππ, π) = ππ
πβ πΎππ3π3
β πΏππ0π3
+
Ξπ0π1
+ π€π1π3. All parameters detailed can be found in [25].
We find that (ππ
β πΈπ)/2 β‘ πΎ
πand (π
π+ πΈπ)/2 β‘ πΏ
π, where πΎ
π
and πΏπare the band structure energies in 2D square lattice of
nearest-neighbor and next-nearest-neighbor transfer, respec-tively. And βπΎ
0β‘ π€ which π€ 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
π=
1
4β
π
(
tanh (βΞ2 + π2β/2π)
βΞ2 + π2β
+
tanh (βΞ2 + π2+/2π)
βΞ2 + π2+
) ,
(9)
where π+
= (ππ
+ πΈπ)/2 + β((π
πβ πΈπ)/2)2
+ πΎ2
0and π
β=
(ππ+πΈπ)/2ββ((π
πβ πΈπ)/2)2
+ πΎ2
0.The π
+and πβrepresent the
upper and lower bands of quasiparticle energy spectra of thehybridization system.We can determine the superconductingcritical temperature π
πby putting Ξ β 0; then
1
π=
1
4β
π
(tanh (π
β/2ππ)
πβ
+tanh (π
+/2ππ)
π+
) . (10)
In the absence of the hybridization interaction, πΎ0
= 0; thatis,
1
π=
1
2β
π
(tanh (π/2π
π0)
π) or 1
π= ln(
2πΎππ·
πππ0
) ,
(11)
where πΎ = 1.78. π = π(0)π, π is the coupling constant, andπ(0) is the constant density of state at the Fermi surface, and
ππ·is theDebye cutoff energy.π
π0is the critical temperature of
superconductor without hybridization that the BCSβs 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(ππ
β πΈπ) and the superconductor with conduction electron
band coexistingwith lower-energy other-electron band (ππ
β«
πΈπ, πΈπ
β 0).
Case 1. The superconductor that the conduction electronband having the same energy as other-electron band.
In this case, the approximation is ππ
β πΈπ. Then, we get
πβ
β ππ
β πΎ0and π+
β ππ
+ πΎ0. The difference of the lower
and upper energy spectra is equal to 2πΎ0. If πΎ0
= 0, theBCSβs 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
π=
1
4(β«
ππ·
βππ·
tanh (πβ/2ππ)
πβ
πππ
+ β«
ππ·
βππ·
tanh (π+/2ππ)
π+
πππ)
β ln(
2πΎβπ2
π·β πΎ2
0
πππ
) .
(12)
The critical temperature is
ππ
= 1.13βπ2
π·β πΎ2
0πβ1/π
. (13)
And the zero-temperature energy gap can be found as
1
π=
1
4β«
ππ·
βππ·
(1
βΞ2 (0) + (ππ
β πΎ0)2
+1
βΞ2 (0) + (ππ
+ πΎ0)2
) πππ
=1
2(sin hβ1 (
ππ·
β πΎ0
Ξ (0)) + sin hβ1 (
ππ·
+ πΎ0
Ξ (0))) .
(14)
For ππ·
β« Ξ(0), we can get
Ξ (0) = 2βπ2
π·β πΎ2
0πβ1/π
. (15)
From (13) and (15), the gap-to-ππratio is obtained:
π =2Ξ (0)
ππ
= 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-π
πratio.
4 Advances in Condensed Matter Physics
To investigate the effect of hybridization and the Debyecutoff on gap-to-π
πratio, we rewrite the gap equation at crit-
ical temperature and at zero-temperature into the form [29]
β«
(ππ·+πΎ0)/(2ππ)
(βππ·+πΎ0)/(2ππ)
tanhπ₯
π₯ππ₯ = β«
2(ππ·+πΎ0)/ππ
β2(ππ·βπΎ0)/ππ
1
βπ 2 + π₯2ππ₯.
(17)
The numerical calculation of (17) is shown in Figure 2.Within the definition of isotope effect coefficient in
harmonic approximation; πΌ = (1/2)(ππ·
/ππ)(πππ/πππ·
) andequation (9), we
πΌ =ππ·
2(tanh (π
β
π·/2ππ) /πβ
π·+ tanh (π
+
π·/2ππ) /π+
π·
tanh (πβ
π·/2ππ) + tanh (π
+
π·/2ππ)
) ,
(18)
where π+
π·= ππ·
+ πΎ0and π
β
π·= ππ·
β πΎ0.
Consider the limiting cases that the hybridization is sosmall with respect to Debye cutoff energy, π
π·β« πΎ0; for π
π·>
2ππ, we can get that πΌ β (π
π·/4)(1/π
β
π·+ 1/π
+
π·) β 1/2, and for
ππ·
< 2ππ, we get πΌ β (π
π·/2)((1/π
π)/(ππ·
/ππ)) β 1/2 that the
BCSβ result.
Case 2. The superconductor that the conduction electronband coexisting with lower-energy other-electron band.
Because of the hybridization Hamiltonian having thesame Greenβs 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 π-electron is associated with themagnetic ordering having lower energy than conductionelectron. We can make the assumption that the π-electronband is at the Fermi level which can be taken as πΈ
πβ 0; then
πβ
βππ
2β πΎ0, π
+β
ππ
2+ πΎ0, for
ππ
2< πΎ0,
πβ
β 0, π+
β ππ, for
ππ
2> πΎ0.
(19)
Substituting above approximation into (10), we can get
1
π=
1
4(β«
ππ·
βππ·
tanh (πβ/2ππ)
πβ
πππ
+ β«
ππ·
βππ·
tanh (π+/2ππ)
π+
πππ)
β ln(2πΎ
πππ
β2ππ·
πΎ0) +
1
2(
ππ·
2ππ
βπΎ0
ππ
) ,
(20)
where,
β«
ππ·
βππ·
tanh (πβ/2ππ)
πβ
πππ
β 2 (ππ·
2ππ
β2πΎ0
2ππ
) + 2 ln(4πΎπΎ0
πππ
) ,
β«
ππ·
βππ·
tanh (π+/2ππ)
π+
πππ
β 2 ln(2πΎππ·
πππ
) .
(21)
The critical temperature is
ππ
= 1.13β2πΎ0ππ·
πβ1/π+(1/2)(ππ·/2ππβπΎ0/ππ). (22)
The gap equation as zero-temperature is
1
π=
1
4β«
ππ·
βππ·
(1
βΞ2 (0) + (πβ)2
+1
βΞ2 (0) + (π+)2
) πππ,
(23)
where,
β«
ππ·
βππ·
1
βΞ2 (0) + (πβ)2
πππ
β (2
Ξ (0)) (ππ·
β 2πΎ0) + 2 sin hβ1 (
2πΎ0
Ξ (0)) ,
β«
ππ·
βππ·
1
βΞ2 (0) + (π+)2
πππ
β 2 sin hβ1 ( ππ·
Ξ (0)) .
(24)
Then, we get
Ξ (0) = 2β2πΎ0ππ·
πβ1/π+(1/2)((ππ·β2πΎ0)/Ξ(0)). (25)
According to (22) and (25), the gap-to-ππratio is
π =2Ξ (0)
ππ
= 3.53π((ππ·β2πΎ0)/ππ)(1/π β1/4). (26)
To investigate the effect of hybridization and the Debyecutoff on gap-to-π
πratio, we rewrite the gap equation at crit-
ical temperature and at zero-temperature into the form [29]
β«
ππ·/ππ
βππ·/ππ
tanh(π¦/4 β β(π¦/4))
π₯ππ₯
= β«
2(ππ·+πΎ0)/ππ
β2(ππ·βπΎ0)/ππ
1
βπ 2 + π₯2ππ₯.
(27)
The numerical calculation of this equation is shown inFigure 2.
Within the definition of isotope effect coefficient in har-monic approximation, (10) and πΈ
πβ 0, we can get
πΌ = (ππ·
2)
Γ (1 + (2π
π/ππ·
) tanh (ππ·
/2ππ)
ππ·
β 2πΎ0
+ 2ππ(tanh (π
π·/2ππ) + tanh (πΎ
0/ππ))
) .
(28)
3. Results and Discussions
We use the hybridized two-band Hamiltonian to investigatethe critical temperature, zero-temperature order parameter,gap-to-π
πratio, 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 5ππ
40
30
20
10
00 1 2 3 4
πΎ0/ππ
Case 1ππ· = 300 π = 0.3Case 1ππ· = 300 π = 0.2Case 1ππ· = 200 π = 0.2
Case 2 ππ· = 200 π = 0.2Case 2 ππ· = 300 π = 0.2Case 2 ππ· = 300 π = 0.3
Figure 1: The ππversus 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 (π
πβ πΈπ) and the
superconductor that conduction electron band coexists withlower-energy other-electron band (π
πβ« πΈπ, πΈπ
β 0). Thecritical temperature, zero-temperature order parameter, gap-to-ππratio, 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 π
π, (Figure 1), we find that the weak-coupling
limit (π < 0.4) can be found in range of ππ·
/ππ
> 10. Then,we investigate the effect of hybridization in weak-couplinglimit with π
π·/ππ
> 10 and πΎ0/ππ
= 0.2, 4.0. We find that ππ
is decreased when the hybridization coefficient increased inCase 2 and no effect in Case 1. In Figure 2, the gap-to-π
πratios
(π ) of Cases 1 and 2 with varied hybridization coefficientsare shown. In Case 1, the π is tended to BCS value, 3.53, asππ·
/ππ
β β. Case 2, π β 3.3β3.9 can be found to dependstrongly on the value of πΎ
0/ππ. The higher value of πΎ
0/ππand
the lower value of π are found to agree with superconductinggapβs 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 (πΌ > 0.5) and less thanBCS value (πΌ < 0.5), but inCase 1we can find only forπΌ > 0.5.However, πΌ in both cases is converse to 0.5 as π
π·/ππ
β β.The Fe-based superconductors are mutiband system
which comprises at least two bands which propose π -waveparing state. The π = 3.68 [31] which shows a consistentmanner with the BCS prediction is found. The value of iso-tope effect coefficient were found to be πΌ β 0.35β0.4 [32] andπΌFe = 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
π
10 12 14 16 18 20ππ·/ππ
Case 2 πΎ0/ππ = 4.0Case 1πΎ0/ππ = 4.0
Case 2 πΎ0/ππ = 0.2Case 1πΎ0/ππ = 0.2
Figure 2: The gap-to-ππratio of Cases 1 and 2 with varied
hybridization coefficients.
πΌ
1.00
0.75
0.50
ππ·/ππ
10 12 14 16 18 20
Case 2 πΎ0/ππ = 4.0Case 1πΎ0/ππ = 4.0
Case 2 πΎ0/ππ = 0.2Case 1πΎ0/ππ = 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-π
π
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 BCSβs results. For the superconductor that conduc-tion electron band coexists with lower-energy other-electronband, the hybridization coefficient show more effect. π
πis
decreased when the hybridization coefficient increases. Wecan get higher and lower value of π and πΌ than BCSβs resultsdepending on the hybridization coefficient. Higher value ofhybridization coefficient, lower value of π , and higher valueof πΌ are found.
Acknowledgments
The financial support of the Office of the Higher EducationCommission, Srinakhariwirout University, and ThEP Centeris acknowledged.
References
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