The Study on Hybridized Two-Band Superconductor

1 Program of Physics and General Science, Faculty of Science and Technology, Rajabhat Nakhon Ratchasima University, Thailand 2 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, Thailand 4Rajamangala University of Technology Phra Nakhon, 399 Samsen Road, Dusit, Bangkok 10300, Thailand


Introduction
Since Moskalenko [1,2] Suhl et al. [3] introduced the twoband model that accounts for multiple energy bands in the vicinity of the Fermi energy contributing electron pairing in superconductor, the two-band model has been applied to high temperature superconductor in copper oxides [4][5][6][7][8][9][10], MgB 2 superconductor [11][12][13], and heavy Fermion superconductor [14,15].Dolgov et al. [16] studied the thermodynamic properties of the two-band superconductor: MgB 2 .The superconducting energy gap, free energy, the entropy, and heat capacity were calculated within the framework of twoband Eliashberg theory.Mazin et al. [17] studied the effect of interband impurity scattering on the critical temperature of two-band superconductor in MgB 2 .Askerzade and Tanatar [18] and Changjan and Udomsamuthirun [19] calculated the critical field of the two-band superconductor by Ginzburg-Landau approach and applied it to Fe-based superconductors.Golubov and Koshelev [20] investigated the two-band superconductor with strong intraband and weak interband electronic scattering rates in the framework of coupled Usadel equation.
The interplay of superconductivity and magnetism is one of the most interesting phenomena of superconductor.The cuprate superconductor exhibits the phase diagram having the magnetic ordered states in the vicinity of the superconducting phase.The antiferromagnetic and ferromagnetic phases are also found in the heavy Fermion superconductor.In ErRh 4 B 6 [21] and HoMo 6 S 8 [22] system, -wave superconductivity shows ferromagnetism in the ground state at intermediate temperature.
The nesting properties of Fermi surface in low dimensional system arise the spin density wave (SDW) state and charge density wave (CDW) state in the interplay of superconductivity and magnetism system.The SDW and the CDW states occurred by Coulomb interaction between electron, and electron-phonon interaction, respectively.Nass et al. [23] used the BCS-type pairing to explain the antiferromagnetic superconductor.Suzumura and Nagi [24] investigated some properties of antiferromagnetic superconductor.They proposed the Hamiltonian of the superconductivity associated with conduction -electron and the antiferromagnetism associated with -electron of rare earth atoms that formed the BCS-type pairing.Ichimura et al. [25] investigated the effect Advances in Condensed Matter Physics of the CDW on BCS superconductor within a mean-field approximation.In this model, the two order parameters of CDW and superconductor were introduced.The tightbinding band in 2D square lattice and nesting vector  = (, ) was used in their calculations.Rout and Das [26] applied the periodic Anderson model (PAM) for calculating the nonmagnetic ground state of heavy Fermion superconductor.The Hamiltonian of the heavy Fermion systems composed of conduction electron band, -electron band, the hybridization of conduction electron and -electron band, BCS-like pairing band, and the intra-atomic Coulomb interaction of -electron.The Hamiltonian was simplified by linearising the intra-atomic Coulomb interaction with the Hartree-Fock approximation then the -electron band energy was   =   +   , where   was bare -electron energy and   was the Coulomb energy of -electron.Finally, their Hamiltonian were consisted of conduction and -electron band including BCS-like pairing band and hybridization term.Panda and Rout [27] studied the interplay of CDW, SDW, and superconductivity in high temperature superconductor in low doping phase.The model of meanfield Hamiltonian including the CDW, SDW and superconductivity was introduced.
In this paper, we modified the hybridized Hamiltonian of Rout and Das [26] to be the two-band superconductor with hybridization.Some physical properties of two-band hybridized superconductor, that is, critical temperature, zerotemperature order parameter, gap-to-  ratio, and isotope effect coefficient, were investigated.

Model and Calculation
According to the hybridized Hamiltonian [26] that consisted of conduction electron and -electron band, BCS-like pairing band, and hybridization term, we have where  +  (  ) and  +  (  ) are the creation (annihilation) operator of conduction electron and -electron.Δ is the superconducting order parameter that Cooper pairs involve only the conduction electron. 0 is the hybridization interaction coefficient of -electron band and conduction band.   =  +    is the intraatomic Coulomb interaction between -electron.They [26] where  0 =   +  − that is the energy collected the noninteraction with a modified -level.
In our model, the two-band superconductor comprise of conduction electron and other-electron band.The superconducting order parameters can occurr by conduction electron and other-electron band.The conduction band makes the intra-atomic Coulomb interaction with other-electron band.We set where  +  (  ) and  +  (  ) are the creation (annihilation) operators of conduction electron and other-electron band.Δ is the superconducting order parameter. 0 is the hybridization interaction coefficient of other-electron band and conduction band.   =  +    is the intra-atomic Coulomb interaction between other-electron.The Hamiltonian is linearised by [26]'s technique; then we get the simplified two-bandhybridized Hamiltonian.We can write the Hamiltonian as where The first Hamiltonian describes the conduction electron Hamiltonian, the second Hamiltonian describes the Hamiltonian of other-electron band, and the third Hamiltonian describes the interact Hamiltonian.Where   and   are the band energies of the conduction electron and other-electron band measured from the Fermi energy.  =  0 + − is the energy collected the non-interaction with a modified other-electron band. +  (  ) and  +  (  ) are the creation (annihilation) operator of conduction electron and other-electron. 0 is the hybridization interaction coefficient of other-electron band and conduction band.Δ is the effective superconducting order parameter occurred by conduction electron and other-electron and assumed to be homogeneous in space.The effective superconducting order parameters is where the -wave like BCS pairing interaction having the same coupling interaction potential is assumed.The effective Advances in Condensed Matter Physics 3 superconducting order parameters are the coupling equation of conduction electron and other-electron that can occurr in magnetic superconductor [23][24][25].However for simplicity of calculation, the same pairing strength is taken [28].
After some calculations, the Green function in Nambu representation is obtained: where   = (2+1),  is temperature,  is an integer, and   and   ( = 1, 2, 3) are the Pauli matrices.Our Green function obtained shows the same form as [25] that investigated the effect of the CDW on BCS superconductor within a meanfield approximation, All parameters detailed can be found in [25].We find that (  −   )/2 ≡   and (  +   )/2 ≡   , where   and   are the band structure energies in 2D square lattice of nearest-neighbor and next-nearest-neighbor transfer, respectively.And − 0 ≡  which  is the order parameter of CDW.This result means that the CDW consideration gives the same result as the hybridization consideration within a mean-field approximation.
From ( 6) and ( 8), the superconducting gap equation is where The  + and  − represent the upper and lower bands of quasiparticle energy spectra of the hybridization system.We can determine the superconducting critical temperature   by putting Δ → 0; then In the absence of the hybridization interaction,  0 = 0; that is, where  = 1.78. = (0),  is the coupling constant, and (0) is the constant density of state at the Fermi surface, and   is the Debye cutoff energy. 0 is the critical temperature of superconductor without hybridization that the BCS's result.Because of the complicated quasi-particle energy spectra obtained, we introduce the two approximated conditions to calculate analytically; the superconductor with conduction electron band having the same energy as other-electron band (  ≈   ) and the superconductor with conduction electron band coexisting with lower-energy other-electron band (  ≫   ,   ≈ 0).Case 1.The superconductor that the conduction electron band having the same energy as other-electron band.
In this case, the approximation is   ≈   .Then, we get  − ≈   −  0 and  + ≈   +  0 .The difference of the lower and upper energy spectra is equal to 2 0 .If  0 = 0, the BCS's superconductor is obtained.In this case, the effect of the hybridization interaction on the BCS superconductor is considered.
We substitute above approximations into (10); then the gap equation becomes The critical temperature is And the zero-temperature energy gap can be found as For   ≫ Δ(0), we can get From ( 13) and ( 15), the gap-to-  ratio is obtained: In this case, we find that the hybridization interaction coefficient decreases the critical temperature and zero-temperature energy gap but has no effect on gap-to-  ratio.
Case 2. The superconductor that the conduction electron band coexisting with lower-energy other-electron band.
Because of the hybridization Hamiltonian having the same Green's function as the charge density wave model, we can apply this model to the superconducting state found in heavy Fermion superconductor.The heavy Fermion superconductor has its origin in the interplay of strong Coulomb repulsion in 4f-and 5f-shells and their hybridizations with the conduction band.The -electron is associated with the magnetic ordering having lower energy than conduction electron.We can make the assumption that the -electron band is at the Fermi level which can be taken as   ≈ 0; then Substituting above approximation into (10), we can get where, The critical temperature is The gap equation as zero-temperature is where, Then, we get According to ( 22) and ( 25), the gap-to-  ratio is To investigate the effect of hybridization and the Debye cutoff on gap-to-  ratio, we rewrite the gap equation at critical temperature and at zero-temperature into the form [29] The numerical calculation of this equation is shown in Figure 2. Within the definition of isotope effect coefficient in harmonic approximation, (10) and   ≈ 0, we can get ) . (28)

Results and Discussions
We use the hybridized two-band Hamiltonian to investigate the critical temperature, zero-temperature order parameter, gap-to-  ratio, and isotope effect coefficient of superconductor.The Green function and gap equation are derived analytically.However, the quasi-particle energy spectra obtained are complicated; then we introduce two approximated cases as, the superconductor that conduction electron band has the same energy as other-electron band (  ≈   ) and the superconductor that conduction electron band coexists with lower-energy other-electron band (  ≫   ,   ≈ 0).The critical temperature, zero-temperature order parameter, gapto-  ratio, 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 numerical calculations of   , (Figure 1), we find that the weak-coupling limit ( < 0.4) can be found in range of   /  > 10.Then, we investigate the effect of hybridization in weak-coupling limit with   /  > 10 and  0 /  = 0.2, 4.0.We find that   is decreased when the hybridization coefficient increased in Case 2 and no effect in Case 1.In Figure 2, the gap-to-  ratios () of Cases 1 and 2 with varied hybridization coefficients are shown.In Case 1, the  is tended to BCS value, 3.53, as   /  → ∞.Case 2,  ≈ 3.3-3.9can be found to depend strongly on the value of  0 /  .The higher value of  0 /  and the lower value of  are found to agree with superconducting gap's behavior of HF system of Rout et al. [30].
The isotope effect coefficient is also investigated and shown in Figure 3.In Case 2, we find that the isotope effect coefficient can be more than the BCS ( > 0.5) and less than BCS value ( < 0.5), but in Case 1 we can find only for  > 0.5.However,  in both cases is converse to 0.5 as   /  → ∞.
The Fe-based superconductors are mutiband system which comprises at least two bands which propose -wave paring state.The  = 3.68 [31] which shows a consistent manner with the BCS prediction is found.The value of isotope effect coefficient were found to be  ≈ 0.35-0.4[32] and  Fe = 0.81 [33].These results indicate that electron-phonon interaction plays some role in the superconducting mechanism by affecting the magnetic properties [31].According to our model, the effective superconducting order parameters

Conclusions
The two-band hybridized superconductor that the pairing occurred by conduction electron band and other-electron band is studied in weak-coupling limit.The formula of critical temperature, zero-temperature order parameter, gap-to-  ratio and isotope effect coefficient are calculated.For the superconductor that conduction electron band has the same energy as other-electron band, the hybridization coefficient Advances in Condensed Matter Physics shows a little effect.The numerical results do not differ much from the BCS's results.For the superconductor that conduction electron band coexists with lower-energy other-electron band, the hybridization coefficient show more effect.  is decreased when the hybridization coefficient increases.We can get higher and lower value of  and  than BCS's results depending on the hybridization coefficient.Higher value of hybridization coefficient, lower value of , and higher value of  are found.